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Physics-Informed Machine Learning in Prognostics and Health Management: A Systematic Literature Review
Christopher Braun, Julian Raible, Marco F. Huber
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Summary
This paper presents a systematic literature review of 212 studies on Physics-Informed Machine Learning (PIML) in Prognostics and Health Management (PHM). It introduces a four-class classification scheme (observational, inductive, learning, and hybrid biases) to categorize how physical knowledge is integrated into ML models. The review finds that PIML improves predictive performance over conventional baselines across various assets, particularly lithium-ion batteries and bearings, though the literature is skewed toward problem-specific solutions. It highlights that while tangible benefits exist, evidence for resolving limitations like generalization and interpretability is insufficient, calling for future work on transferable design patterns and uncertainty-aware models.
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Relation Signals (8)
Physics-Informed Machine Learning â appliedto â Prognostics and Health Management
confidence 95% ¡ This work investigates how PIML is being leveraged in the context of PHM through a systematic literature review
Physics-Informed Machine Learning â classifiedby â Observational Bias
confidence 92% ¡ The review introduces a four-class classification scheme, consisting of observational bias...
Physics-Informed Machine Learning â classifiedby â Inductive Bias
confidence 92% ¡ The review introduces a four-class classification scheme, consisting of ... inductive bias...
Physics-Informed Machine Learning â classifiedby â Learning Bias
confidence 92% ¡ The review introduces a four-class classification scheme, consisting of ... learning bias...
Literature on PIML â focuseson â Lithium-ion batteries
confidence 90% ¡ the literature is heavily skewed toward lithium-ion batteries and bearings
Literature on PIML â focuseson â Bearings
confidence 90% ¡ the literature is heavily skewed toward lithium-ion batteries and bearings
Physics-Informed Machine Learning â mitigates â limitations of purely data-driven models
confidence 90% ¡ Physics-Informed Machine Learning (PIML) helps mitigate these limitations by incorporating prior physical knowledge directly into the ML pipeline
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Abstract
Abstract:In modern industry, keeping complex systems reliable, safe, and efficient hinges on Prognostics and Health Management (PHM). Machine Learning (ML) has largely driven advancements in diagnostics and prognostics, yet purely data-driven models face inherent limitations, such as poor generalization, an inability to infer causal relationships, and a lack of interpretability. Physics-Informed Machine Learning (PIML) helps mitigate these limitations by incorporating prior physical knowledge directly into the ML pipeline, thereby fostering growing interest in its application to PHM. This work investigates how PIML is being leveraged in the context of PHM through a systematic literature review of 212 studies. The review introduces a four-class classification scheme, consisting of observational bias, inductive bias, learning bias, and hybrid approaches, and further categorizes studies by PHM task. Across all four classes, the reviewed studies consistently demonstrate improved predictive performance over conventional baselines across a broad range of assets, although the literature is heavily skewed toward lithium-ion batteries and bearings, and dominated by problem-specific solutions. Overall, the review indicates that physics-informed approaches already provide tangible benefits, whereas claims of improvements concerning some of the aforementioned limitations lack sufficient supporting evidence. Future research should prioritize transferable design patterns, benchmarks comparing integration strategies, and uncertainty-aware models that are lightweight and robust enough for online deployment in real-world settings.
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Physics-Informed Machine Learning in Prognostics and Health Management: A Systematic Literature Review Christopher Braun 1,2*â , Julian Raible 1,2â and Marco F. Huber 1,2 1 Institute of Industrial Manufacturing and Management IFF, University of Stuttgart, Allmandring 35, Stuttgart, 70569, Baden-W Ěurttemberg, Germany. 2 Fraunhofer Institute for Manufacturing Engineering and Automation IPA, NobelstraĂe 12, Stuttgart, 70569, Baden-W Ěurttemberg, Germany. *Corresponding author(s). E-mail(s): christopher.braun@iff.uni-stuttgart.de; Contributing authors: julian.raible@iff.uni-stuttgart.de; marco.huber@ieee.org; â These authors contributed equally to this work. Abstract In modern industry, keeping complex systems reliable, safe, and efficient hinges on Prognostics and Health Management (PHM). Machine Learning (ML) has largely driven advancements in diagnostics and prognostics, yet purely data-driven models face inherent limitations, such as poor generalization, an inability to infer causal relationships, and a lack of interpretability. Physics-Informed Machine Learning (PIML) helps mitigate these limitations by incorporating prior physical knowledge directly into the ML pipeline, thereby fostering growing interest in its application to PHM. This work inves- tigates how PIML is being leveraged in the context of PHM through a systematic literature review of 212 studies. The review introduces a four-class classification scheme, consisting of observational bias, inductive bias, learning bias, and hybrid approaches, and further categorizes studies by PHM task. Across all four classes, the reviewed studies consistently demonstrate improved predictive per- formance over conventional baselines across a broad range of assets, although the literature is heavily skewed toward lithium-ion batteries and bearings, and dominated by problem-specific solutions. Over- all, the review indicates that physics-informed approaches already provide tangible benefits, whereas claims of improvements concerning some of the aforementioned limitations lack sufficient supporting evidence. Future research should prioritize transferable design patterns, benchmarks comparing inte- gration strategies, and uncertainty-aware models that are lightweight and robust enough for online deployment in real-world settings. Keywords:Systematic literature review, Physics-informed machine learning, Prognostics and health management, Prior physical knowledge, Hybrid approaches The version of record of this article, first published in Journal of Intelligent Manufacturing, is available online at Publisherâs website: https://dx.doi.org/10.1007/s10845-026-02930-3. This arXiv version is content-equivalent to the version of record but differs in four respects: (i) the list of studies excluded after full-text analysis, available as supplementary material (referred to as Online Resource 1 in the version of record), is included here as an appendix; (i) citations are consistently disambiguated, resolving cases in which several distinct references share an identical in-text citation string; (i) section headings are numbered, so that the numbered cross-references used throughout the text can be resolved; and (iv) all figures are embedded such that the text they contain remains selectable and searchable. No claims, results, or conclusions have been altered. Please cite the version of record. 1 arXiv:2608.10047v1 [cs.LG] 10 Aug 2026 Abbreviations (Technical Terms) AEAutoencoder BiLSTMBidirectional Long Short-Term Memory BPFIBall Pass Frequency Inner Race BPFOBall Pass Frequency Outer Race BSFBall Spin Frequency CNNConvolutional Neural Network CWTContinuous Wavelet Transform DLDeep Learning DOFDegrees of Freedom ECAEfficient Channel Attention ECMEquivalent Circuit Model EKFExtended Kalman Filter FACFrequency-Aware Convolution FEFinite Element FLOPFloating Point Operation GANGenerative Adversarial Network GNNGraph Neural Network GPGaussian Process GPRGaussian Process Regression GRUGated Recurrent Unit HIHealth Index IMLInformed Machine Learning kNNK-Nearest Neighbor LSTMLong Short-Term Memory MLMachine Learning MLPMultilayer Perceptron NNNeural Network ODEOrdinary Differential Equation PDEPartial Differential Equation PdMPredictive Maintenance PFParticle Filter PHMPrognostics and Health Management PIMLPhysics-Informed Machine Learning PINNPhysics-Informed Neural Network ReLURectified Linear Unit ResNetResidual Network RFRandom Forest RFRRandom Forest Regression RULRemaining Useful Life RLReinforcement Learning RNNRecurrent Neural Network SEISolid Electrolyte Interphase SOCState of Charge SOHState of Health SPMSingle-Particle Model SVRSupport Vector Regression SVMSupport Vector Machine TGDSTheory-Guided Data Science TLTransfer Learning TRLTechnology Readiness Level 1 Introduction Prognostics and Health Management (PHM) has emerged as a cornerstone of modern industrial operations, driven by the increasing need for relia- bility, safety, and efficiency in complex engineering systems (Vogl, Weiss, & Helu, 2019). Specifically, PHM provides the methodological foundation for Predictive Maintenance (PdM), enabling organi- zations to transition from reactive or schedule- based strategies through condition-based monitor- ing to predictive measures, supporting informed decision-making (C. Huang et al., 2024). This transition has been accelerated by advances in sensing technologies, connectivity, and industrial digitalizationâcentral pillars of the Industry 4.0 paradigm. As industrial assets become more inter- connected and operational demands intensify, PHM plays a critical role in reducing unplanned downtime, optimizing maintenance costs, and ensuring continuous production. Machine Learning (ML) has significantly influ- enced PHM research, with its adoption growing substantially in recent years (Sajjadi, Dinmo- hammadi, & Shafiee, 2025). ML models have demonstrated strong performance in tasks such as anomaly detection (Cannizzaro et al., 2025), fault diagnosis (Y. Zhang, Zhang, & Liu, 2025), and Remaining Useful Life (RUL) prediction (Y. Yin, Tian, & Liu, 2025). Despite these achievements, purely data-driven approaches face inherent limi- tations, particularly when deployed in real-world industrial settings. In such settings, sensor mea- surements are frequently noisy, incomplete, or inconsistent, operational conditions vary widely across assets and environments, and failure data remain sparse due to the rarity of catastrophic events. Moreover, ML models often lack inter- pretability and struggle to extrapolate beyond the 2 state space covered by the training data (Hag- meyer, Zeiler, & Huber, 2022). These limita- tions are particularly critical in industrial sectors where reliability, safety, and trustworthiness are paramount, driving the need for more advanced modeling strategies. Physics-InformedMachineLearning (PIML) (Karniadakis et al., 2021) constitutes a promising paradigm for addressing these chal- lenges by incorporating prior physical knowledge into the ML pipeline. By combining the scala- bility and predictive capabilities of ML with the structure and interpretability of physics-based modeling, PIML offers a pathway toward robust, generalizable, and physically consistent PHM solutions. These characteristics align strongly with the stringent reliability and transparency demands of industrial settings, where decisions based on model outputs often carry significant operational or safety implications (Zio, 2022). Beyondmethodologicalmotivations,the industrial context itself amplifies the relevance of PIML. Modern industrial systems operate under harsh, dynamic, and heterogeneous conditions. Assets may experience variable loads, nonlinear degradation, or rapid transitions between oper- ating regimes. Sensor availability and quality can differ across machines and sites, and data sharing is frequently impeded by privacy and security concerns. Fleet-level variability intro- duces further complexity. Models must generalize across equipment units that share design prin- ciples but exhibit different usage patterns or environmental exposures, often driving costly model reengineering (Zeng, Zhang, Lang, Wang, & Chen, 2025). These practical challenges under- score the need for models that do not rely solely on data, but instead leverage physical insights to ensure robustness and adaptabilityâprecisely the strengths that physics-informed approaches promise to provide. Notably, the growing adop- tion of PIML extends beyond PHM into adjacent fields such as structural health monitoring (Rizvi & Abbas, 2023), underscoring its cross-domain relevance for systems subject to degradation. Despite the growing body of research at the intersection of PIML and PHM, the field remains fragmented across ways of integrating physics, across PHM tasks, and across appli- cation domains. Existing surveys tend to focus on specific assets, such as lithium-ion batter- ies (Meng & Li, 2019), gas turbines (Farhat & Altarawneh, 2025), or bridges (Mammeri, Barros, Conde-Carnero, & Riveiro, 2025). Moreover, they tend to adopt a more exploratory approach, pro- viding detailed insights in specific areas while leav- ing some aspects less systematically addressed. Given that PIML began attracting significant attention following the seminal work by Karni- adakis et al. (2021), the field has been developing rapidly, making it challenging for recent surveys to fully capture the current state of research. As a result, researchers lack a unified classification that enables systematic comparison of how physics is integrated across different PHM tasks and appli- cation domains. Practitioners, in turn, have no consolidated evidence base to guide the selection of an appropriate integration strategy for a given industrial setting. This limits both the cumulative advancement of methods and their translation into operational practice. At its core, this work addresses the question of how PIML is being leveraged in the context of PHM, and what the key challenges and oppor- tunities are. To answer this systematically and ensure both reproducibility and completeness, the most comprehensive systematic literature review of PIML in PHM to date is conducted, cover- ing 212 studies. The outcome provides researchers, developers, and practitioners with the necessary insights to advance data-driven PHM applications by effectively incorporating prior physical knowl- edge. Specifically, the research questions this work seeks to answer are: 1.Knowledge(a) What types of prior physical knowledge are being leveraged, and (b) what forms of representation are employed? 2.Incorporation(a) How can prior physical knowledge be incorporated, and (b) how does the form of representation influence which approaches to incorporation are feasible? 3.Practice(a) How does incorporating prior physical knowledge help overcome limitations of purely data-driven methods, and (b) what are the primary challenges in developing and applying physics-informed approaches? The remainder of this work is organized as follows. Section 2 provides background on PHM, 3 physics-informed learning, and a brief descrip- tion of related work. Section 3 describes the methodology underlying the systematic literature review, including the search strategy, as well as the screening and quality assessment procedures. Section 4 presents the four-class classification scheme, including observational bias, inductive bias, learning bias, and hybrid approaches. Sub- sequently, the identified studies are summarized. Section 5 analyzes methodological trends and dis- cusses current challenges and opportunities for future advancements in industrial PIML-based PHM. Finally, Section 6 concludes with a syn- thesis of key insights and implications, concisely answering the outlined research questions. 2 Background The following section outlines the theoretical foun- dation necessary to understand the key concepts in this review. PHM is introduced first, covering its core tasks, as well as various approaches to its implementation. Among these, hybrid approaches are particularly promising, with PIML at the fore- front. This paradigm is defined and explored, along with related research fields. Finally, an overview of related work is provided to position this review within the broader literature. 2.1 Prognostics and Health Management PHM is an engineering discipline that focuses on detecting, isolating, and diagnosing potential faults in a system, assessing its current State of Health (SOH), and predicting its RUL with the aim of preventing unplanned downtimes, enhanc- ing reliability, and supporting additional system- level objectives. These diagnostic and prognostic measures play a crucial role in ensuring contin- uous operation by monitoring system health and predicting incipient faults before they progress into catastrophic failures. Hence, PHM not only improves operational efficiency but also enhances the safety of the monitored system. However, implementing PHM presents several challenges that vary depending on the chosen approach for modeling the system. While the adoption of PHM in industrial settings holds significant potential, it also requires addressing various technical and Table 1Key strengths and limitations of physical model-based and purely data-driven approaches (Baur, Albertelli, & Monno, 2020; Karniadakis et al., 2021). Physical Model- Based Approaches Purely Data-Driven Approaches Strengths ⢠Accurate and reliable ⢠Interpretable ⢠Robust ⢠Generalizable ⢠Scalable ⢠Low implementation effort Limitations ⢠Requires complete system knowledge ⢠High implemen- tation effort ⢠Highly system-specific ⢠Relies on representative and sufficient data ⢠Less interpretable ⢠Poor extrapolation capability ⢠Risk of implausible predictions organizational obstacles to fully realize its ben- efits. With PHM encompassing both diagnostics and prognostics, four essential tasks are typically delineated for implementation (Hagmeyer et al., 2022; Jia, Huang, Feng, Cai, & Lee, 2018): 1.Fault detectionis aimed at determining the presence or absence of a fault. 2.Diagnosisis aimed at attributing observed faults to their root causes. 3.Health assessmentis aimed at estimating the systemâs current SOH or risk of failure. 4.Prognosisis aimed at predicting the future development of the SOH or RUL. To carry out these tasks, a range of method- ological approaches is used, varying in the extent to which they rely on physical knowledge, data- driven insights, or a combination of both. These approaches are generally subdivided into three dis- tinct categories (Atamuradov, Medjaher, Dersin, Lamoureux, & Zerhouni, 2017; N.-H. Kim, An, & Choi, 2017): ⢠Physical model-based approaches ⢠Purely data-driven approaches ⢠Hybrid approaches According to Gouriveau, Medjaher, and Zer- houni (2016), physical model-based approaches 4 ârequire the construction of a dynamic model rep- resenting the behavior of the system and integrat- ing the degradation mechanism (mainly by models of fatigue, wear, or corrosion), whose evolution is modeled by a deterministic law or by a stochas- tic process.â In contrast, purely data-driven approaches (including statistical and ML meth- ods) leverage monitoring dataâeither directly or via extracted featuresâto model a systemâs behavior and health state (Goodman, Hofmeister, & Szidarovszky, 2019). Hybrid approaches, which integrate aspects of the aforementioned methods, seek to leverage their strengths while mitigating their individual limitations (see Tab. 1). Combin- ing the accuracy and robustness of physical models with the flexibility and adaptability of data-driven models enables enhancing the overall performance of the respective solution. The aim is to capital- ize on the synergies between these approaches, ultimately providing a more comprehensive and effective solution that can accommodate diverse operational scenarios. Nonetheless, this does not preclude scenarios in which a physics-based or data-driven approach is preferable. Definitions of hybrid approaches may vary, however, particularly in terms of the extent to which physics is included. Certain hybrid approaches employ complete physical models, while others rely on prior physical knowledge that may be insufficient for holistic modeling. The choice of approach is generally dictated by the underlying physics of the problem, the availabil- ity of relevant data, and specific requirements imposed by the intended solution. The continuous automation of modern industrial machinery leads to increasingly complex degradation processes, which are often poorly understood, dynamic, and highly nonlinear (Zio, 2022). As a conse- quence, high-fidelity physics-based modeling is becoming increasingly difficult, if not impossible. Accordingly, the integration of (partial) physical knowledge into data-driven methods is gaining momentum. 2.2 Physics-Informed Machine Learning The incorporation of prior knowledge alongside empirical data constitutes a central paradigm in ML research (see Fig. 1). von Rueden et al. (2021) provide a general conceptualization of learning from such hybrid information sources, thereby establishing the foundational framework of Informed Machine Learning (IML). According to this framework, prior knowledge is expected to originate from independent sources, be formally represented, and be explicitly integrated into the learning process. Karpatne et al. (2017) propose a related framework with a specific focus on sci- entific knowledge, referred to as Theory-Guided Data Science (TGDS). Narrowing the focus fur- ther, Karniadakis et al. (2021) introduce PIML as a means to improve the modeling of physical systems by embedding prior physical knowledge directly into ML models. Instead of relying solely on data, PIML incorporates physics such as governing differen- tial equations, conservation laws, or symmetries to ensure that model predictions remain physi- cally consistent, which is especially valuable in regimes where data are sparse or noisy (Kar- niadakis et al., 2021). This makes PIML par- ticularly attractive in scientific and engineering domains, where high-fidelity measurements can be expensive or limited by experimental feasibility. Operationally, PIML can be implemented through a variety of complementary strategies, includ- ing data-centric approaches, design-level inter- ventions, and regularization-based constraints. Yet developing PIML remains a nontrivial task, involving challenges such as balancing data fidelity with physics constraints, managing computational costs, and ensuring efficient training (Jahani- Nasab & Bijarchi, 2024). Nonetheless, PIML has become a prominent paradigm for developing models that are both data-driven and physically informed, balancing accuracy, interpretability, and robustness. Three main pathways have been defined for embedding physics into ML models, following the principles outlined by Karniadakis et al. (2021). Each is characterized by the introduction of an appropriate bias: ⢠Observational biascan be introduced directly through the training data. ⢠Inductive biascan be introduced by tai- lored interventions to the model design. ⢠Learning biascan be introduced through modifications to the learning algorithm. 5 Informed Machine Learning (IML) von Rueden et al., 2021 IML âdescribes learning from a hybrid information source that consists of data and prior knowledge.â Theory-Guided Data Science (TGDS) Karpatne et al., 2017 TGDS âaims to leverage the wealth of scientific knowledge for improving the effectiveness of data science models in enabling scientific discovery.â Physics-Informed Machine Learning (PIML) Karniadakis et al., 2021 PIML is defined as âthe process by which prior knowledge stemming from our observational, empirical, physical or mathematical understanding of the world can be leveraged to improve the performance of a learning algorithm.â KnowledgeGeneralSpecific Fig. 1IML (von Rueden et al., 2021) provides a foundational framework for integrating various forms of prior knowledge into ML. Within this context, TGDS (Karpatne et al., 2017) focuses specifically on incorporating scientific knowledge, while PIML (Karniadakis et al., 2021) narrows the scope even further to leverage prior physical knowledge. The hypothesis space provides a useful lens for understanding how these biases affect learn- ing. It denotes the set of all functions a model could, in principle, choose to map inputs to out- puts; for example, all functions realizable by a Neural Network (N) with a given architecture. Observational bias, stemming from the training data, does not alter the space itself. It merely biases the training process toward functions that better fit the observed data, leaving the set of representable functions intact. Inductive bias, in contrast, acts directly on the model design and explicitly shapes the hypothesis space. Embed- ding symmetry constraints, conservation laws, or other physical principles into the model restricts the hypothesis space to functions consistent with these principles, actively enforcing physical plausi- bility. Learning bias exerts a more subtle influence: choices in optimization algorithms, regularization, or training strategies such as early stopping guide the search toward certain solutions, making some regions more likely to be explored while leaving the space itself unchanged. In summary, observa- tional bias influences the selection of hypotheses within the existing space (not imposing con- straints), inductive bias modifies the structure of the space itself (imposing hard constraints), and learning bias guides the learning process toward particular regions, thereby effectively prioritiz- ing certain solutions over others (imposing soft constraints). 2.3 Related Work Extensive reviews have emerged around both PHM (Atamuradov et al., 2017; Hu, Miao, Si, Pan, & Zio, 2022; Tsui, Chen, Zhou, Hai, & Wang, 2015; Zio, 2022) and PIML (S. Cai, Mao, Wang, Yin, & Karniadakis, 2021; Cuomo et al., 2022; Karniadakis et al., 2021) individually. However, while both fields have advanced significantly, the synergy between them is increasingly recognized as crucial for addressing the intricate challenges of effectively managing system health. Accord- ingly, several reviews have sought to synthesize the state of research at the intersection of PHM and PIML, among which the following were known to the authors prior to conducting this systematic literature review: W. Deng, Nguyen, Medjaher, Gogu, and Morio (2023), Kundu, Darpe, and Kulkarni (2020), and the review by Meng and Li (2019)âall of which were also identified through the systematic approach employed in this work. Hence, a detailed description is omitted here, since these (along with eight additional reviews) will be examined in detail in Section 4.2. Furthermore, a broader perspective on PIML in the context of intelligent manufacturing is provided by Leng et al. (2026). 3 Methodology The following section describes the methodology employed to conduct the systematic literature 6 Table 2Keywords used to identify the initial set of records. A wildcard operator (*) accounts for morphological variations. CategoryKeywords Prior Physical Knowledge (âphysics-inform*â OR âphysics-infus*â OR âphysics-guid*â OR âphysics-induc*â OR âphysics-constrain*â OR âphysics-promot*â OR âphysics-enhanc*â OR âphysics- inherit*â OR âphysics-awareâ OR âphysics-basedâ OR âtheory-guid*â OR âalgebraic equation*â OR âdifferential equation*â OR âODE*â OR âPDE*â OR âconservation law*â OR âspatial invariance*â OR âprobabilistic relation*â OR âsimulat* dataâ OR âprior knowledgeâ) AND Machine Learning (âartificial intelligenceâ OR âpattern recognitionâ OR âdata-drivenâ OR âmachine learningâ OR âdeep learningâ OR âscientific machine learningâ OR âhybrid machine learningâ OR âsupervised learningâ OR âunsupervised learningâ OR âself-supervised learningâ OR âreinforcement learningâ OR âactive learningâ OR âautomated learningâ OR âmeta-learningâ OR âtransfer learningâ OR âfew-shot learningâ OR âmulti- task learningâ OR âensemble learningâ OR âprobabilistic modelingâ OR âsequence modelingâ OR âtime series forecastingâ OR âMarkov model*â OR âhidden Markov modelâ OR âGaussian process*â OR âsupport vector machine*â OR âsupport vector regressionâ OR âdecision tree*â OR ârandom forest*â OR âk-nearest neighborsâ OR âk-means clusteringâ OR âhierarchical clusteringâ OR âGaussian mixture modelâ OR âgradient boostingâ OR âboosting algorithmsâ OR âAdaBoostâ OR âXGBoostâ OR âLightGBMâ OR âCatBoostâ OR âneural network*â OR âlong short-term memoryâ OR âgated recurrent unitâ OR âautoencoderâ OR âgenerative adversarial network*â OR âBayesian network*â OR âtransformer*â OR âfoundation modelâ OR âCNN*â OR âRNN*â OR âLSTM*â OR âpretext taskâ OR âdownstream taskâ OR âQ-learningâ OR âdeep Q-networkâ OR âpolicy gradient methodsâ OR âactor-critic methodsâ OR âmodel-free learningâ OR âmodel-based learningâ) AND Prognostics and Health Management (âprognostics and health managementâ OR âPHMâ OR âdiagnos*â OR âprognos*â OR âhealth assessmentâ OR âhealth managementâ OR âhealth monitoringâ OR âcondi- tion monitoringâ OR âpredictive maintenanceâ OR âremaining useful lifeâ OR âRULâ OR âstate of healthâ OR âend of lifeâ OR âdegrad*â OR âfault isolationâ OR âfault detect*â OR âanomaly detect*â OR âdamage detect*â OR âfailure detect*â OR âfail- ure predictionâ) AND NOT Healthcare(âhealthcareâ OR âmedic*â OR âclinic*â OR ânursingâ OR âcancer*â OR âillness*â OR âdisease*â OR âsicknessâ) review on PIML in PHM. Starting from the research questions formulated in Section 1, the process encompasses the selection of relevant key- words and databases, a rigorous screening proce- dure to identify all relevant studies, and a quality- based refinement to distill the final selection. By systematically analyzing the literature, this work provides a solid foundation for synthesizing cur- rent physics-informed approaches to diagnostics and prognostics, identifying methodological gaps, and guiding future research directions. 3.1 Keywords The keywords used to identify the initial set of records are listed in Table 2. In total, 105 unique terms were defined across four categories:prior physical knowledge,machine learning,prognostics and health management, andhealthcare. The first three categories ensure relevance to the scope of this work. The last category serves as an implicit filter to rule out irrelevant healthcare-related records resulting from ambiguities in PHM-related keywords. 7 Relevant keywords were identified by examin- ing the research questionsâ core concepts and cor- responding synonyms. This also included screen- ing prominent studies explicitly addressing the scope of this review for their author-assigned keywords, employing large language models to generate additional keyword suggestions, and con- sulting peers to review and validate the keyword collection. Furthermore, keywords were iteratively refined by performing exploratory searches in the selected databases, enabling the identification and exclusion of keywords of lower importance or higher ambiguity (e.g., âenergyâ or âforceâ within the category ofprior physical knowledge; âfaultâ or âfailureâ within the category ofprognostics and health management). This approach ensured that the final keyword set was both precise and comprehensive. 3.2 Databases Scopus and Web of Science (WoS) were selected as the primary databases due to their comprehensive coverage of peer-reviewed literature across vari- ous disciplines, ensuring a robust and thorough search. Additionally, the preprint server arXiv was included to capture the most recent research developments. This decision was made to provide a more accurate depiction of the current state of research, acknowledging that many cutting-edge studies are first disseminated through preprints before formal publication, especially in the realm of ML. TGDS, a precursor to PIML, was formally established around 2017 (as stated in Sec. 2.2), marking a significant milestone in integrating sci- entific knowledge with ML techniques. However, the search covered the period from 2012 onwardâ a year widely recognized within the ML commu- nity as a pivotal moment due to the breakthrough results of AlexNet (Krizhevsky, Sutskever, & Hin- ton, 2012) in the ImageNet competition. This choice reflects the consensus that 2012 represents the inception of modern ML and Deep Learning (DL) (Mienye & Swart, 2024). Yet the search on arXiv was limited to the period from 2023 to the present, under the assumption that high-quality studies submitted to arXiv before 2023 would have already been published in peer-reviewed journals or conference proceedings. The complete search string was constructed by connecting the keywords of the various cat- egories with the appropriate logical operators (see Tab. 2). The syntax for the logical opera- tors was adapted to the specific requirements of each database. The fields examined for relevant records included the title, abstract, and author- assigned keywords. For arXiv, which does not provide searching capabilities for author-assigned keywords, only the title and abstract were con- sidered. All keywords were enclosed in quotation marks in order to ensure exact phrases in the search queries. Moreover, both hyphenated and non-hyphenated variants of keywords were auto- matically retrieved by the databases. 3.3 Screening Procedure The screening procedure was aided by ASRe- view (Van De Schoot et al., 2021), which is an ML-based tool designed to streamline the sys- tematic review process. It leverages active learn- ing algorithms to prioritize the most relevant records from a large corpus, significantly reduc- ing the manual effort required for initial screening. Records are presented iteratively, with their order continually updated based on reviewer feedback (researcher-in-the-loop) to facilitate the identifica- tion of relevant records. Additionally, ASReview supports bias-free screening with respect to author names, affiliations, and other details by providing only the title and abstract of each record. ASReview was employed to manage the sub- stantial number of records initially retrieved from the searched databases. A structured approach to screening was implemented, consisting of the following steps: 1. Constructing a dataset containing the records to screen. 2. Specifying inclusion and exclusion criteria to determine whether records are considered relevant. 3. Defining a stopping criterion to determine at which point screening will conclude. 4. Conducting screening in two cycles: 4.1 Configuring the active learning algorithm and providing a subset of records for its initial training. 8 4.2 Performing the screening based on the specified criteria until the stopping crite- rion is met. 5. Extracting the records identified as relevant to compile the subset of records for further analysis. The set of records initially retrieved from the designated databases constitutes the dataset, with all duplicates removed. Only records containing both a title and an abstract are retained. To deter- mine the relevance of identified records, inclusion and exclusion criteria are specified. The inclu- sion criterion requires that a record substantively addresses all three areas: prior physical knowl- edge, ML, and PHM. While the search string (see Tab. 2) ensures that matching terms appear in the title, abstract, or author-assigned keywords of every retrieved record, syntactic matching alone is insufficient to guarantee genuine topi- cal relevanceârecords may, for instance, reference related concepts in a negating or merely periph- eral context. The reviewer therefore objectively assesses whether all three areas are adequately addressed in substance, rather than relying solely on keyword occurrence. Regarding the exclusion criterion, a record is excluded if the studied sys- tem, component, material, or process is not inves- tigated with respect to degradation during opera- tion, as diagnostics and prognostics fundamentally rely on the assetâs current health state. The scope of eligible assets further excludes transportation systems and their infrastructure, unmanned aerial vehicles, consumer electronics, and other technical systems used in residential settings. Van De Schoot et al. (2021) state that, based on their simulation studies, reviewing 8â33 % of the total number of records using ASReview is typically sufficient to identify 95 % of the relevant records. Based on these findings, the stopping cri- terion is defined as follows: at least 8 % of the records must be reviewed, and screening is termi- nated at the latest once 33 % have been screened. Within these bounds, screening is considered com- plete when 50 consecutive irrelevant records are encounteredâa heuristically set threshold. Screening comprises two cycles, with the first cycle leveraging a rather simple but fast con- figuration of the active learning algorithm. The minimum requirement for labeled training data is to include at least one record labeled as rele- vant and one as irrelevant. The aim during this cycle is to gather all relevant records that are more easily distinguishable from the rest of the dataset. At a certain point, however, this configu- ration reaches its limitations, and inevitably, the stopping criterion is met. Nevertheless, it can be assumed that the dataset still contains relevant records, though identifying them will require a more nuanced approach. This necessitates a sec- ond cycle of screening, where a more sophisticated model is employed to capture all remaining rel- evant records that are more difficult to detect. These records often elude initial screening due to subtle semantic nuances and complex contextual variations that a simpler model might overlook. The result (i.e., all labeled records) of the first cycle serves as the initial training data for the more complex model. Upon reaching the stopping criterion for the second time, it is assumed that a representative subset of relevant records has been identified. Two reviewers screened records independently in alternating one-hour sessions. Records of debat- able relevance were discussed and decided upon through mutual consultation. Periodic discussions ensured a shared understanding and consistent application of the criteria outlined earlier. 3.4 Quality-Based Refinement After the screening procedure is completed, the collection of identified records is refined by retain- ing only those that meet specified quality criteria. With respect to journal articles, the quality- based refinement depends on both the impact factor (taken from the Journal Citation Reports provided by Clarivate Analytics (2024)) and the SCImago Journal Rank (SJR) (provided by SCImago (2024)). The former is required to be 3 or greater, while the SJR is required to be Q2 or better. If a journal is assigned to multiple categories within the SJR, it must maintain a min- imum ranking of Q2 across all categories. Both the impact factor and SJR are referenced accord- ing to the publication year of the article, with the most recent available values used when the corre- sponding yearâs metrics are not yet released. An article will still be considered if only one of the two metrics is available, provided the journal meets the required standard. Articles from journals for 9 Table 3Conferences considered for the quality-based refinement of conference papers (in alphabetical order). AcronymConference CMSCIRP Conference on Manufacturing Systems ESRELEuropean Safety and Reliability Conference ETFAIEEE International Conference on Emerging Technologies and Factory Automation ICPHMIEEE International Conference on Prognostics and Health Management INDINIEEE International Conference on Industrial Informatics PHMSCAnnual Conference of the PHM Society (including its counterparts in Europe and Asia) RAMSInternational Conference on Reliability and Maintainability SMCIEEE International Conference on Systems, Man and Cybernetics which neither metric is available are excluded from the final selection. For conference papers, no established metric or ranking system with broad interdisciplinary applicability comparable to those used for jour- nals exists. Consequently, the quality-based refine- ment for these records is applied differently. Only papers presented at conferences recognized for their relevance and impact in the field of PHM are considered (see Tab. 3). Contributions from the preprint server arXiv cannot undergo quality-based refinement. Accord- ingly, arXiv papers are incorporated into the final selection of relevant records without additional assessment if they are identified as relevant during the screening procedure. In contrast, other forms of publication, such as book chapters or technical reports, are excluded due to the lack of a reli- able strategy for assessing their academic rigor and impact. 3.5 Results The methodology was applied twice (referred to as two rounds), with database searches conducted on each occasion to systematically gather rele- vant records and provide an up-to-date depiction of the state of research. This approach revealed a notable increase in publications over time, reflect- ing heightened research activity and growing inter- est at the intersection of PIML and PHM. Among the sources, Scopus yielded the highest number of records, followed by WoS, while arXiv produced the fewest due to the more restricted time frame applied to that search. While the period from January 2012 to August 27, 2024, yielded 6,586 records, the subsequent period from August 27, 2024, to May 4, 2025, produced 1,874 records. This indicates a marked acceleration in contribu- tions, with nearly 30 % of the previous 12.5 yearsâ output occurring within this brief interval. After retrieval, these records were preprocessed, result- ing in a reduction to 3,956 and 1,026 records, respectively. Thereafter, systematic screening and filtering yielded the final set, which formed the basis for analyzing the most current and relevant literature on PIML in PHM (see Fig. 2). The two rounds comprised a total of three cycles, each utilizing a specifically configured active learning algorithm within the ASReview framework. In the first cycle of round one, a comparatively simple yet computationally efficient configuration was adopted. Feature extraction was performed usingterm frequency-inverse document frequency, and a Naive Bayes classifier served as the underlying model for prioritizing records. The query strategy and the balancing strategy were set tomixedanddynamic resampling, respectively. Upon reaching the stopping criterion, a more sophisticated configuration was introduced for the second cycle. Specifically, feature extraction was switched tosBERT, which produces contextually richer embeddings, and the classifier was replaced by an N, while the query and balancing strate- gies remained unchanged. The second round of screening (cycle three) adopted the same advanced configuration as cycle two, leveraging all previ- ously labeled records from the first round as initial training data. In preparation for the first cycle, 20 labeled records were provided for initialization of the active learning algorithm. Although the minimum requirement is only two labeled records, this pro- vided the model with a more informative starting point. Records initially labeled as relevant were not guaranteed inclusion in the final review, as 10 some may have been excluded during quality- based refinement or full-text analysis. The initial labeled set comprised the following records: ⢠Ten relevant records: Badora, Bartosik, Graziano, and Szolc (2023); Y. Chen, Rao, Feng, and Zuo (2022); Ellis, Heyns, and Schmidt (2022); Gareev et al. (2021); Garpelli, Alves, Cavalca, and de Castro (2023); Gurgen and Dinh (2022); L. Ma, Tian, Zhang, Guo, and Hu (2024); H. Sun, Cao, Zhao, and Kang (2018); Yucesan and Viana (2019); Q. Zhou and Tang (2023) ⢠Ten irrelevant records: J. Chen, Tang, Rak- stad, Patrick, and Zhou (2020); J. Gao, Zheng, and Yang (2021); Gong et al. (2022); He et al. (2024); Hu et al. (2017); McMahon et al. (2024); S. Pan, Li, Zeng, Guo, and Hu (2019); Sajedi, Eltouny, and Liang (2023); Wanasundara, Wickramas- inghe, Schaubroeck, and Muthukumarana (2023); Zjavka, MiËs Ěak, and Prokop (2017) Two reviewers assessed a total of 1,189 (30.1 %) and 339 (33.0 %) records for relevance during the two rounds of screening, respectively (see Tab. 4). Screening was performed in an alternating fashion, requiring 43 iterations over- all. During round one (i.e., cycle one and two), screening was terminated upon encountering 50 consecutive irrelevant records, in accordance with the stopping criterion, whereas round two con- cluded automatically upon reaching the maximum screening threshold of 33 %. The difference in the total number of records screened per reviewer can be attributed to several factors, such as encoun- tering longer abstracts, varying complexity in determining whether all three areas (prior physi- cal knowledge, ML, and PHM) were substantively addressed, and varying cognitive load. The screen- ing yielded a total of 384 records across both rounds, a number considered excessive even for a thorough review. The application of the quality-based refine- ment effectively reduced the number of records. The final set comprised 212 records, including 180 journal articles, 18 conference papers (ESREL, ICPHM, PHMSC, and RAMS), and 14 preprints sourced from arXiv. Accordingly, this work con- stitutes the most comprehensive review to date in Retrieval Round 1 August 27, 2024 Scopus: N= 3,812 WoS: N= 2,614 arXiv: N= 160 In total:N= 6,586 Round 2 May 4, 2025 Scopus: N= 926 WoS: N= 858 arXiv: N= 90 In total:N= 1,874 Records retrieved: Screening Quality Filtering Full-text Analysis N= 3,956N= 1,026 Records included after deduplication: N= 1,189N= 339 Records screened: N= 269N= 115 Records identified as relevant: N= 143N= 69 Records included after quality filtering: N= 78N= 51 Records included after full-text analysis: Fig. 2This flowchart illustrates the key stages of the methodology and the corresponding number of records at each stage for both rounds. this area of research. All 212 records were subse- quently subjected to full-text analysis, after which 83 records were excluded due to insufficient align- ment with the scope of this review. All exclusions were transparently documented with their cor- responding rationale (see Appendix). Hence, 129 records will be presented and discussed in this review. 11 Table 4This table provides a detailed overview of the screening results across all three cycles, where the first two authors correspond to reviewer A and B (in no particular order). Round 1Round 2 Cycle 1Cycle 2In total (Round 1) Cycle 3In total (Round 1+2) Iterations181735843 Records screened61357611893391528 Reviewer A372346718174892 Reviewer B241230471165636 Relevant records149120269115384 Irrelevant records4644569202241144 3.6 Limitations Given that a reproducible systematic literature review depends on transparency, all potential limitations related to the outlined methodology are clearly articulated, mainly concerning the aided screening procedure and the subsequent quality-based refinement. Screening a large num- ber of records required incorporating an addi- tional toolâASReviewâaimed at facilitating the identification of those that are actually relevant. Although ASReview relies on iterative reviewer feedback, semantic nuances may still pose chal- lenges for the underlying active learning algo- rithms. As a result, some relevant records may not have been surfaced by the algorithm before the stopping criterion was met. This constitutes an inherent limitation of active learning-based screening that cannot be fully eliminated without exhaustive manual review of the entire corpus of nearly 5,000 records. However, the stopping crite- rion, defined in accordance with Van De Schoot et al. (2021), inherently implies that approximately 5 % of relevant studies may remain unidentified by design. The decision to apply a quality-based refine- ment was partly driven by the substantial number of records identified as relevant during screening. The set of 384 records was considered impractical for full review, necessitating a reduction to a more manageable number. Retaining only high-quality studies not only reduced the number of records but also enhanced both the relevance of the find- ings and the reliability of the conclusions drawn. Nonetheless, the thresholds for the impact factor and SJR were established based on the authorsâ expert judgment, as was the selection of high- impact conferences, which may be considered a limitation of the review. 4 Literature Review The following section provides a synthesis of all studies included in the systematic literature review, classified according to their methodologi- cal approach to combining physics and ML. The classification scheme adopts and further refines the three pathways outlined by Karniadakis et al. (2021), imposing a more stringent framework. Additionally, hybrid approaches are recognized as a distinct class, with the rationale discussed in detail below. After outlining the classification scheme, a summary of the included review studies is presented, emphasizing the motivation for con- ducting the current review. This is followed by a detailed depiction of the current state of research based on the remaining studies. 4.1 Classification Approaches to implementing PHM are generally subdivided into model-based, data-driven, and hybrid approaches (see Sec. 2.1). In this context, PIML is generally regarded as a paradigm within hybrid approaches. Nonetheless, a key observa- tion motivated distinguishing PIML from hybrid approaches, resulting in a classification scheme comprising four classes: observational bias, induc- tive bias, learning bias, and hybrid approaches (see Fig. 3). While all four approaches involve physics and ML, the last category is conceptually dis- tinct with respect to the role that physics plays within the overall model. Crucially, it departs from the core principle of IML that prior knowledge is explicitly integrated into the ML pipeline (von Rueden et al., 2021), since hybrid approaches couple two independent models. In light of this, studies in the fourth class could technically be con- sidered outside the scope of this review. However, 12 Physics-Informed Machine Learning Hybrid Approaches In-series coupling In-parallel coupling Training Data Model Design Learning Algorithm Observational Bias (no constraints) Represents physics via additional observations, indirectly influencing training while leaving the hypothesis space unchanged. Hypothesis SpaceML Pipeline Inductive Bias (hard constraints) Learning Bias (soft constraints) Enforces physics by design, explicitly constraining the hypothesis space to functions consistent with the embedded prior knowledge. Imposes physics on the learning process, actively steering the model toward favorable regions of the hypothesis space. Pred. PML P Pred. ML Pred. P ML Fig. 3The classification scheme underlying this review comprises four classes, with the first three classes representing genuine physics-informed approaches and the fourth capturing hybrid approaches. to foster understanding of these conceptual differ- ences and to provide a comprehensive overview of the intersection between physics and ML in PHM, this class has been deliberately included. In doing so, the distinct role of PIML within the broader landscape becomes more evident. The first three classes are based on the well-established three pathways for PIML. While inductive and learning bias are defined sufficiently to allow accurate classification, observational bias requires a stricter interpretation to avoid con- flating physics-informed approaches with conven- tional ML. In the original work, Karniadakis et al. (2021) describe two ways of introducing observational bias, namely âthrough data that embody the underlying physics or carefully crafted data augmentation procedures.â While the for- mer applies to virtually any ML problem involving real-world systems whose behavior follows phys- ical principles, the latter provides little practical guidance, as âcarefully craftedâ is not further defined. To resolve the former ambiguity, the def- inition of prior knowledge as formulated by von Rueden et al. (2021) is adopted, which speci- fies that it âexist[s] in an external, separated way from the learning problem and the usual training data.â Hence, observational bias is not introduced by the mere use of empirical data obtained from physical systems. As a result, the inclusion of additional (physics-informed) data becomes obligatory to incorporate this form of bias. Primary approaches include the use of sim- ulated data, either to enrich the training data or as the source domain data within a Transfer Learning (TL) settingâboth strategies essentially addressing data scarcity. This also implies, how- ever, that training (and testing) exclusively on simulated data does not qualify as PIML, as it fails to meet the requirement that prior knowledge is separate from the training data. The ambiguity surrounding carefully crafted data augmentation procedures is resolved as fol- lows. On the one hand, it is virtually impossible to determine what qualifies as carefully crafted, which is why conventional data augmentation techniques are categorically excluded from the scope of PIML. By the same reasoning, feature engineering (regardless of its complexity) does not constitute PIML either. Ultimately, such mea- sures remain conventional ML in that they operate on or derive from existing training data. On the other hand, the integration of entire physics-based 13 models in conjunction with an ML model forms the basis of the fourth class. While some studies employing a hybrid approach could, in princi- ple, have been classified under observational bias according to the broad definition given by Kar- niadakis et al. (2021), it was established earlier that they violate the requirement of explicit inte- gration of prior knowledge into the ML pipeline. Instead, the physics-based and ML components interact largely independently. Based on a thorough examination of stud- ies in this fourth class, two types of approaches emerged, referred to as in-parallel and in-series, respectively. Drawing on terminology from elec- trical engineering, these labels aptly capture the relationships between the physics-based and ML models. Primary in-parallel approaches include residual learning and ensemble methods. In-series approaches involve integrating the physics-based and ML models sequentially, so that the out- put of one model directly informs the input of the other. Conceptually, this sequential integra- tion can occur in two variants, with the ML model directing the physics-based model, or vice versa. Importantly, these hybrid approaches dif- fer in two key aspects with respect to observa- tional bias and feature engineering, respectively. Unlike approaches that incorporate observational bias, in which physics-based simulators are used only offline and are no longer required once the ML model is trained, hybrid approaches retain a physics-based model as an integral component of the prediction pipeline during both training and inference. Additionally, they go beyond fea- ture engineering, where fixed algebraic relations are used to transform input variables but do not constitute a model with meaningful physi- cal parameters. Hence, studies following a hybrid approach involve a physics-based component cor- responding to a parameterized model, combined either in parallel or in series with the ML model. In addition, a clear distinction is drawn with respect to inductive-bias approaches that appear to employ an in-series coupling. If a physics-based model is inserted as a fixed, differentiable module within the computational graph (i.e., the model output is passed through the physics-based model prior to computing the loss, and gradients are propagated through this module), it is referred to as being a part of the ML model. This links back to the hypothesis space: all admissible predictions are, by construction, consistent with the employed physics module. All studies (excluding reviews) are classified and presented following the classification scheme outlined above. Each class is further subdivided by the four PHM tasks: fault detection, diagnosis, health assessment, and prognosis (see Sec. 2.1). Within these task-specific sections, studies are grouped and reported according to related use cases, where possible. Studies that incorporate multiple mechanisms for embedding physics into ML, or that target multiple PHM tasks, are addressed as follows. If mechanisms or tasks are clearly separable, they are reported individually and may therefore appear multiple times across the following sections. Studies with tightly cou- pled mechanisms are assigned to the class that best reflects the primary driving mechanism. The same logic applies to PHM tasks. For each class of approaches, a representative example is pre- sented in a dedicated figure, highlighting the prior knowledge employed and its incorporation. The schematics illustrating each method are adapted from the corresponding study and simplified to highlight the essential information, where blue denotes the ML components and orange the physics components. Lastly, the Sankey diagram in Figure 4 offers a visual overview, capturing overall trends regarding the approaches employed for specific PHM tasks across various assets, thereby offering context for subsequent analysis of methodological patterns. 4.2 Reviews The relevant literature comprises eleven review studies published between 2019 and 2025, explor- ing how the integration of physics into data-driven methods has advanced the broader field of PHM. These reviews offer valuable insights into the fieldâs evolution and present unique perspectives. The following summaries are presented in order of relevance to the current work, highlighting each reviewâs contributions, methodologies, and individual strengths and limitations. W. Deng et al. (2023) present a broad overview of PIML in PHM, without focusing on any spe- cific domain or task. The review is well-motivated, highlighting the advantages of PIML over physi- cal model-based and purely data-driven methods, respectively. While the methodology is the most 14 PIMLâ PHMâ Use Case Observational Bias (25) Inductive Bias (37) Learning Bias (43) Hybrid Approaches (20) Fault Detection (7) Diagnosis (25) Health Assessment (50) Prognosis (43) Lithium-ion Battery (37) Other (33) Cutting Tools (9) Pump (7) Turbofan Engine (7) Metal Specimen (6) Bearing (29) ApproachPHM TaskAsset Fig. 4This Sankey diagram shows three categoriesâapproach,PHM taskandassetâeach containing multiple items, with connections between them representing the state of research based on the 118 studies discussed in Sections 4.3â4.6. The width of each path connecting two items is proportional to the number of studies supporting that link, reflecting the relative strength of evidence. To highlight areas of focus, paths supported by five or more studies across all three categories (i.e., studies that employ the sameapproach, target the samePHM taskand address the sameasset) are shown in darker gray. Within theassetcategory, only those studied by five or more contributions are represented as individual items, while all remaining assets are aggregated under âOtherâ to preserve visual clarity. Moreover, the number of studies corresponding to each item is indicated in brackets. As some studies contribute to multiple items within a category (e.g., by employing different approaches), category totals may exceed 118. detailed among the reviews discussed here, it lacks specifics on the screening procedure, only stat- ing that it was done manually without explaining the criteria used to differentiate relevant from irrelevant studies. The identified studies are orga- nized in a manner reminiscent of the three path- ways outlined by Karniadakis et al. (2021), albeit using different terminology. However, a significant concern is that, despite identifying 122 relevant studies, only 69 are categorized into the three approaches, leaving 53 unaddressed without any explanation from the authors. The comprehensive review by Y. Wu et al. (2024) focuses on the application of PIML to anomaly detection and condition monitoring, but fails to establish a connection to PHM. With the methodology only briefly outlined, key details on screening and selection are missing. Supported by informative figures, the various strategies for integrating physics into ML provide a clear struc- ture for the review. Overall, it offers a valuable contribution with a thorough analysis of recent developments, though the level of detail can be excessive. 15 Table 5Overview of all identified reviews, listed in order of relevance. Dashes indicate omitted information. ReferencePublication Year No. of Studies Time Period W. Deng et al. (2023) 20231222013â2023 Y. Wu, Sicard, and Gadsden (2024) 2024107â H. Li, Zhang, Li, and Si (2024) 20241062003â2022 Khan, Yairi, Tsutsumi, and Nakasuka (2024) 2024â R. Yan et al. (2025) 2025â Fassi, Heiries, Boutet, and Boisseau (2024b) 2024â Zhao et al. (2024) 2024â Cuesta, Leturiondo, Vidal, and Pozo (2025) 2025872018â2024 S.-P. Zhu et al. (2023) 2023â Kundu et al. (2020) 2020â Meng and Li (2019) 2019â2009â2018 H. Li et al. (2024) thoroughly review methods for predicting RUL, focusing on PIML while also identifying and discussing the fusion of physics- based and data-driven models and the devel- opment of stochastic degradation models. The review demonstrates technical depth and includes striking figures to clarify various approaches, but the repeated subdivision of PIML methods with inconsistent terminology impedes comprehension. Although the methodology for identifying relevant literature is briefly outlined, the authors them- selves acknowledge that an exhaustive collection of studies remains elusive. Khan et al. (2024) review physics-based learn- ing for system health management but fail to establish a clear connection to PHM. Although the study provides a transparent background, the methodology is vaguely described, offering only examples of databases and keywords and failing to detail the screening procedure. The authors refer- ence Karniadakis et al. (2021) when introducing PIML, yet merely distinguish between physics- based loss functions and âvarious architecturesââ a category that lacks clear definition and fails to meaningfully differentiate between the diverse approaches emerging in the field. Rather than offering a comprehensive analysis of the identi- fied literature, the review discusses some PIML approaches in general terms, with only a limited selection of examples provided. The review by R. Yan et al. (2025) sets out to introduce a âuniversal concept, knowledge driven machine learning, for integrating diverse knowl- edge into machine learning pipeline [sic] in PHM domain.â However, it largely reproduces the IML taxonomy by von Rueden et al. (2021) without demonstrable novelty. While the authors attempt to synthesize their findings systematically and provide tangible case studies, the articleâs con- tribution is weakened by conceptual ambiguity, methodological omissions, and linguistic inaccu- racies, limiting its value as a rigorous or original synthesis of the field. Fassi et al. (2024b) provide a comprehensive overview of PdM for power converters, structured around model-based, data-driven, and hybrid approaches, with notable emphasis on PIML. However, the review overlooks PHM as the foun- dational framework for PdM and follows a narra- tive rather than systematic methodology. More- over, the discussion of studies employing PIML partly diverges from power converters, drawing substantially on adjacent domains. Lastly, the absence of an in-depth discussion of the reviewed studies limits broader insights, leading to a brief and generic outlook on future work. S.-P. Zhu et al. (2023) provide a brief review of the application of PIML in structural integrity, including failure mechanism modeling and PHM. However, the reviewâs completeness is undermined by the lack of a reported methodology. Instead of offering a critical analysis, it mainly reports the identified studies, missing opportunities to extract valuable insights, such as the prior physi- cal knowledge used, resulting in a weak foundation for discussing current challenges. The overview of ML-based battery safety by Zhao et al. (2024) serves as a solid entry point, outlining the core mechanisms driving battery faults and failures. Yet it provides no information 16 on the methodology, rendering the review narra- tive rather than systematic and leaving its com- pleteness uncertain. Despite the title emphasizing prognostics, studies spanning the entire PHM spectrum are covered. With respect to PIML, the structure is somewhat diffuse: ML in conjunction with battery models is first presented (mainly as a source of domain-specific features), yet several of these approaches effectively fall under PIML, with a later section specifically dedicated to PIML further blurring conceptual boundaries. The review by Cuesta et al. (2025) provides a comprehensive overview of PHM in wind energy, focusing on RUL estimation for key turbine com- ponents such as gearboxes, generators, blades, and bearings. It outlines three main degradation modeling approaches: physics-based, data-driven, and hybrid models. In discussing hybrid models, the authors emphasize the integration of physi- cal knowledge into ML frameworks but do not explicitly situate this discussion within the emerg- ing field of PIML. The review further identifies challenges related to uncertainty quantification, integration of physical knowledge, environmental variability, and system complexity. However, the absence of a clear methodology for study selec- tion raises concerns about the completeness and representativeness of the reviewed literature. Kundu et al. (2020) present a review of diag- nostic and prognostic approaches for gears. Offer- ing both technical depth and clarity, this review is an excellent starting point for researchers and practitioners developing or applying PHM in this area. However, the review lacks information on the process used to gather the relevant literature. Additionally, the authors report only a few stud- ies on hybrid approaches, explaining that few such methods have been developed, and attempt to cover all major approaches up to and including 2020ânone of which employ PIML. The review on PHM of lithium-ion batteries by Meng and Li (2019) is structured into physics- based, data-driven, and hybrid approaches, with the latter focusing primarily on Particle Fil- ter (PF)-based and Kalman filter-based methods. Given the lack of methodological rigor, the work constitutes a descriptive survey rather than a sys- tematic analysis. Moreover, the absence of studies employing PIML as a tool for PHM, likely due to the reviewâs 2019 publication date, underscores the rapid progress of the field. While acknowledging the significant contribu- tions of previous reviews at the intersection of PHM and PIML, it is concluded that a more comprehensive and systematic review is needed to fully address the research questions outlined in Section 1. Although these reviews provide valuable insights, they often offer only partial coverage, focusing on particular domains, e.g., mechanical components (Kundu et al., 2020), batteries (Meng & Li, 2019; Zhao et al., 2024), power convert- ers (Fassi et al., 2024b), or wind energy (Cuesta et al., 2025), or specific PHM tasks, such as RUL prediction (H. Li et al., 2024), and in cer- tain instances, extending more broadly to related areas, such as condition monitoring (Y. Wu et al., 2024) or PdM (Fassi et al., 2024b). More- over, in some cases, PIML is not the central focus but is instead addressed only as a periph- eral topic. Furthermore, a recurring issue across all reviews is the lack of a systematic approach and detailed descriptions, which undermines sci- entific rigor, as transparency and reproducibility are essential to research integrity. Notably, most reviews fail to report the number of included stud- ies or the time period they cover (see Tab. 5). In response, this review seeks to bridge existing gaps and advance the understanding and application of PIML in PHM by providing a comprehensive, focused exploration of the state of the art, guided by a rigorous methodology. 4.3 Observational Bias Observational bias is characterized by introduc- ing prior physical knowledge directly through the training data. The identified studies are organized by PHM task in Tables 6â8. A representative example is illustrated in Figure 5. 4.3.1 Fault Detection No studies have been identified that introduce observational bias for the purpose of fault detec- tion. 4.3.2 Diagnosis A total of ten studies employing observational bias for diagnosis have been identified (see Tab. 6). X. Liu et al. (2024) aim to solve the problem of high similarity among different faults in lithium- ion battery voltage signatures. Considering faults, 17 A New Dynamic Model and Transfer Learning Based Intelligent Fault Diagnosis Framework for Rolling Element Bearings Race Faults: Solving the Small Sample Problem Dong et al., 2022 Use Case:Bearing fault diagnosis Prior Knowledge:4-DOF dynamic model â˘Representable as a nonlinear mass-spring-damper model: â˘= [, , , , = (,) â˘The state vector collects inner/outer race displacements â˘The system matrices are diagonal mass-damping-stiffness terms, and the forcing term groups nonlinear Hertzian contact forces, gravity, imbalance and localized race faults Incorporation:Simulation-driven TL â˘Pretraining on a simulated vibration dataset, followed by fine- tuning on limited real data â˘For this, different parameter transfer strategies are employed: freeze (fr), fine-tune (ft), and random initialization (ra) 1D-CNN Feature Extractor Classifier Head Simulation Data Experimental Data Source Domain Target Domain 1D-CNN Feature Extractor Classifier Head Transferring Parameters Parameter Transfer Strategies fr fr ft ft fr ft ra ft ft ra fr ra ft fr ra fr ++++++++ Feature Extractor Classifier Head Fig. 5A representative example of observational-bias approaches is proposed by Y. Dong, Li, Zheng, Wang, and Xu (2022), demonstrating how prior physical knowledge can be introduced as additional training data. Own illustration based on the corresponding study. See the original work for full technical details. such as internal short circuits, capacity anomaly, and State of Charge (SOC) anomaly occurring in a series-connected battery system, the authors calculate fault features by applying a mean differ- ence model to terminal voltages. These serve as input to a vision Transformer, followed by a Mul- tilayer Perceptron (MLP) head for fault diagnosis. They use a discretized first-order Equivalent Cir- cuit Model (ECM) to generate simulation data for pretraining the model, while fine-tuning is per- formed on experimental data, leading to improved accuracy. Y. Qin et al. (2024) focus on the challenge of accurately diagnosing bearing faults given imbal- anced fault samples. An inverse Physics-Informed Neural Network (PINN) is employed to esti- mate the parameters of a 2-Degrees of Free- dom (DOF) dynamic bearing model, specifically stiffness and damping ratio, from real vibration data. This enables the generation of simulated fault data whose frequency-domain characteris- tics closely match those of real measurements. The simulated data are then used to supple- ment imbalanced datasets, significantly enhancing the accuracy of a Convolutional Neural Network (CNN)-based diagnostic model for bearing fault detection. In another work, Y. Qin et al. (2025) address the challenge of unseen compound faults in bearing fault diagnosis. The approach employs a composed model consisting of a CNN-based feature extractor, a Cycle-Generative Adversarial Network (GAN)-based semantic mapping model (trained on simulated data) and a Cycle-GAN- based feature generator (trained on measured single-fault data), and a multi-agent deep Rein- forcement Learning (RL) model for the final diagnosis. Simulated vibration signals for various single and compound fault types are generated with a 2-DOF bearing dynamic model. These signals are employed in conjunction with their respective envelope spectra as semantics to train the semantic mapping model. Hence, zero-shot learning regarding real-world compound faults can be performed. The proposed approach demon- strates superior performance in comparison to other state-of-the-art methods for three bear- ing datasets. Y. Dong et al. (2022) present a fault diagnosis framework for bearing race faults that addresses the small-sample problem using a dynamic model and TL strategies (see Fig. 5). Simulated data generated from a 4-DOF bear- ing dynamic model are used to pretrain a CNN, and parameter transfer strategies (i.e., selectively freezing or fine-tuning different parts of the net- work) adapt the model to limited real-world data. By aligning simulation and real-world conditions, the approach enables effective feature transfer and reduces distribution mismatch. Compared 18 Table 6All studies employingobservational biasto addressdiagnosis, listed in alphabetical order. Observational Bias for Diagnosis ReferencePrior Physical KnowledgeUse Case TypeRepresentation Y. Dong et al. (2022)4-DOF dynamic bearing model ODE systemBearing Y. Li et al. (2024)Numerical simulation of bear- ing fault vibration, based on impulse-response signal model Algebraic equationBearing X. Liu et al. (2024)Discretized first-order ECMRecursive algebraic equation Lithium-ion bat- tery Z. Ma, Fu, Xu, and Zhang (2025) Multi-DOF rotor-bearing dynamic model Coupled nonlinear ODEs Bearing Pettorossi, Heiries, Ger- ard, and Rosini (2024) Proton exchange membrane fuel cell simulation ODE systemFuel cell Y. Qin, Liu, Wang, and Mao (2024) 2-DOF dynamic bearing model ODE systemBearing Y. Qin, Liu, Li, and Mao (2025) 2-DOF dynamic bearing model ODE systemBearing Song, Xiong, Zhong, Xiao, and Tang (2022) Rigid-flexible multibody dynamics simulation of a plane- tary gearbox Differential-algebraic equation system Planetary gearbox Y. Zhang, He, Tang, Ren, and Xiang (2024) System level axial piston pump model (rotor, bearings, fluid), bearing dynamic model ODE systemAxial piston pump, bearing Y. Zhu et al. (2025)4-DOF dynamic bearing model ODE systemBearing to conventional ML models (i.e., Support Vec- tor Machine (SVM), K-Nearest Neighbor (kNN), Random Forest (RF), and MLP), both with and without simulation-based TL, the proposed CNN- based TL approach consistently outperforms all alternatives in bearing fault diagnosis under small- sample conditions. Y. Li et al. (2024) address the challenge of bearing fault diagnosis with unlabeled data. Both completely unlabeled real data and labeled simulation data from a bear- ing failure simulation are combined in a semi- supervised method. The simulation covers time- domain vibration signals for faults occurring in the outer race, inner race, and rolling elements. Specifically, a multi-kernel kNN graph connect- ing both data sources is built and edge weights are refined with both layer attention and dot- product attention. The resulting node embeddings are fed into fully connected layers for fault clas- sification. The proposed approach outperforms advanced Graph Neural Network (GNN)-based methods. The domain gap problem regarding sim- ulation and real data in bearing small-sample fault diagnosis is studied by Y. Zhu et al. (2025). A bearing dynamic model is used to generate time-series vibration data and corresponding fault labels, which are used as inputs for the gener- ator of a conditional deep convolutional GAN instead of random noise. The discriminator is trained to distinguish real vibration data from GAN-generated data, encouraging the generator to produce synthetic signals that both respect the underlying fault mechanism and closely match the distribution of real measurements. This generated data can then be used to train a fault diag- nostic model alongside real data. The presented method achieves higher accuracy and lower vari- ance compared to a scenario in which the original simulation and real data are used directly to train the fault diagnostic model. Z. Ma et al. (2025) address few-shot bearing fault diagnosis by using a nonlinear rotor-bearing system model to simulate mechanistic vibration signals for different fault types and severities. These signals are combined with equipment-specific characteristics extracted from healthy data, denoised via wavelet packet decomposition, and further refined using a cosine- similarity-guided noise injection, yielding a mixed 19 dataset that comprises simulated and scarce real fault samples. Across multiple DL architectures, training on the mixed dataset results in superior diagnostic accuracy compared to training solely on measured, simulated, or GAN-generated data. The small-sample problem, resulting from the limited availability of fault data for mechani- cal components, is addressed by Y. Zhang et al. (2024) through the use of simulation mod- els. Time-series data from the simulation are used as the source domain, along with unlabeled experimental data as the target domain, to train an adversarial domain adaptation model with a feature extractor, healthy pattern recognizer, and domain discriminator. Experiments on both axial piston pump and bearing datasets demon- strate that the proposed method significantly enhances fault-diagnosis accuracy and generaliza- tion under small-sample conditions, outperform- ing other domain-adaptation methods. Song et al. (2022) present a method for plan- etary gearbox fault diagnosis based on TL. In this study, gearbox dynamics are simulated to generate labeled training data, addressing the challenge of limited real-world fault samples. A deep TL framework is then applied to gener- alize the fault detection model across different operating conditions. The framework employs a CNN-based feature extractor with shared param- eters for both simulated source domain data and experimental target domain data, as well as a classifier. Besides the classification loss, a domain adaptation loss is employed to guide domain- invariant features. Experimental results for differ- ent TL tasks demonstrate the effectiveness of this approach in identifying cracked gear and miss- ing tooth faults. The proposed method performs slightly better than comparable TL baselines. Both the small-sample problem present in pro- ton exchange membrane fuel cell fault diagnosis and the distribution mismatch between simulated and real data are studied by Pettorossi et al. (2024). They use a supervised domain-adversarial adaptation model, combining a Long Short-Term Memory (LSTM)-based feature extractor, a fully connected fault classifier, and a gradient-reversal- based domain discriminator. A simulated dataset produced by a calibrated physics-based proton exchange membrane fuel cell model is used as the source domain, whereas measurements from a real stack serve as the target domain. In experi- ments incorporating varying amounts of real data into the training process, the proposed approach consistently outperforms a baseline trained exclu- sively on real data without the domain discrim- inator. Notably, these performance gains become more pronounced as the quantity of real data decreases. 4.3.3 Health Assessment As shown in Table 7, ten studies employ obser- vational bias for health assessment. The study by Y. Liu et al. (2024) focuses on accurately esti- mating the SOH of lithium-ion batteries under realistic fast-charging conditions using minimal labeled data. A reduced-order electrochemical model is calibrated on laboratory cycling tests, expanded via stochastic perturbation of aging parameters, and then used to generate partial- charge profiles for pretraining a CNN. The pre- trained network is subsequently adapted to spe- cific batteries via TL with only a small number of real partial-charge segments. Ablation stud- ies (pre- and post-TL, varying real-data volume and distribution) demonstrate that incorporating simulated data significantly improves generaliza- tion and enables extrapolation from early-life data to mid- and late-life degradation states. Kohtz et al. (2022) introduce a multi-fidelity framework for estimating the SOH of lithium-ion batteries from a single short partial charging segment. A 1D Finite Element (FE) Solid Electrolyte Inter- phase (SEI)-growth model is first used to simulate capacity fade and SEI thickness, which are sub- sequently fused with experimental data to train a co-kriging multi-fidelity model that links SEI thickness to capacity (SOH). Separately, nested Gaussian Process Regression (GPR) models are trained on FE data to map operating conditions and SEI levels to voltage points and charging time, which are then used to infer SEI thicknessâand thus SOHâfrom a single partial charging segment without any prior usage history. The authors val- idate their proposed framework by comparing the FE model, single-fidelity surrogates, and the co- kriging multi-fidelity model, with the latter attain- ing the smallest capacity estimation errors. How- ever, it is not quantitatively benchmarked against alternative SOH estimation methods. Thelen et al. (2022) address the online health assessment of 20 Table 7All studies employingobservational biasto addresshealth assessment, listed in alphabetical order. Observational Bias for Health Assessment ReferencePrior Physical KnowledgeUse Case TypeRepresentation Bachar and Bortman (2024) Gear vibration modelNonlinear second-order ODE Gear Kohtz, Xu, Zheng, and Wang (2022) 1D FE SEI-growth modelCoupled PDEsLithium-ion bat- tery S. Li, Lin, Shi, Shi, and Zhu (2024) Empirical flank wear evolution model, mechanistic milling force model Algebraic equationsMilling Y. Liu, Zhou, Pang, Fan, and Zhang (2024) Extended SPMNonlinear state- space system derived from coupled PDE- based electrochemical equations Lithium-ion bat- tery Matania et al. (2023)Gear vibration modelNonlinear second-order ODE Gear Mei, Chen, Yang, and Zhai (2024) Physics-of-failure-based relay degradation model Coupled first-order ODEs with time- varying stochastic input parameters, eval- uated via FE-based numerical simulation Electromagnetic relays Navidi, Thelen, Li, and Hu (2024) Half-cell degradation modelAlgebraic equationsLithium-ion bat- tery Ren et al. (2025)Electromagnetic and electrome- chanical model Coupled ODEs imple- mented via 2D FE simulation Brushless direct- current motor H. Sun et al. (2022)Ultrasonic guided-wave propa- gation and scattering model PDEs solved via FE simulation Microcrack quan- tification in aluminum plate Thelen et al. (2022)Half-cell degradation modelAlgebraic equationsLithium-ion bat- tery lithium-ion batteries by estimating both capac- ity (SOH) and three internal degradation modes. Prior physical knowledge is incorporated via a half-cell model, which is used to generate degrada- tion data and subsequently combined with limited early-life experimental data. Two approaches are compared: data augmentation, in which simulated and experimental data are merged to train a single model, and delta learning, in which an estimator trained on simulated data is corrected by a sec- ond model trained on experimental data. Across both scenarios, data augmentation consistently outperforms delta learning for all four lightweight ML models tested, with the elastic net (a linear regression model with combined L1/L2 regulariza- tion) achieving the highest overall performance. As a follow-up study, Navidi et al. (2024) compare four approaches, including the elastic net-based data augmentation and delta learning methods from the previous study (Thelen et al., 2022). Additionally, delta learning employing GPR is examined, alongside another method that intro- duces a learning bias (see Sec. 4.5.3). Among the approaches incorporating observational bias, the GPR-based method yields the lowest errors for capacity estimation. A qualitative comparison of all four methods, considering aspects such as model flexibility, data requirements, and ease of 21 implementation, provides a broader perspective on their applicability. S. Li et al. (2024) propose a method for online tool condition monitoring in milling, where the goal is to estimate flank wear from cutting- force signals under varying operating conditions. A mechanistic tool wear model and a milling force model are first calibrated on a small set of offline measurements and then used to generate synthetic data. Simulated force signals are used to pretrain a DL model that combines a residual network with learnable soft-threshold attention (for denois- ing and feature extraction) and a Bidirectional Long Short-Term Memory (BiLSTM) (for map- ping force features to simulated wear labels). With a strong initialization provided by pretraining on simulated data, the model is subsequently fine- tuned on a limited set of real samples. In rigorous comparisons with its purely physics-based and purely data-driven counterparts, as well as with PIML approaches and several standard ML base- lines, the proposed method consistently achieves lower errors and more robust wear estimation across multiple milling conditions, while requiring substantially fewer real labels. By quantifying existing microcracks from ultrasonic guided-wave measurements obtained via a specially designed, unidirectionally focus- ing electromagnetic acoustic transducer and a circular array of receivers around the suspected defect location, H. Sun et al. (2022) tackle the health assessment of metallic plate structures. To compensate for the small set of experimen- tal measurements, synthetic guided-wave signals are generated using an FE simulation that spans a broad range of crack geometries. The architec- ture includes two branches with shared weights that process experimental and simulated signals in parallel, enabling the model to compare them and infer crack length. The method additionally incorporates a learning bias (see Sec. 4.5.3). While the proposed model achieves substantially lower errors compared to conventional N variants, the study does not include ablation studies to infer the corresponding contribution of the various biases incorporated. Matania et al. (2023) address fault severity estimation for spur gears when only one faulty experimental sample is available. A dynamic gear model is used to generate vibration sig- nals. The vibration data undergo several pre- processing steps, such as angular resampling, synchronous averaging, propagation through an estimated transfer function, and extraction of time-domain features. The authors then mix the preprocessed simulation data with real data to serve as input for a kNN regressor. The ablation studies conducted do not specifically study the effect of incorporating simulation data or compar- isons against state-of-the-art baselines. Building on this work, some of the authors propose an unsupervised framework with a broader scope that can perform health assessment for multiple fault types (Bachar & Bortman, 2024). In their study on health assessment of brush- less direct-current motor stators, Ren et al. (2025) use a stator damage matrix derived from torque, speed, and load signals, which is then mapped to a scalar Health Index (HI). To this end, an LSTM is trained on a mixed dataset of experimental and FE-simulated torque sequences to predict a dam- age matrix describing turn-level open-circuit and short-circuit faults. Based on the predicted dam- age matrix, a scalar HI is computed via cosine similarity between the current damage state and a health reference state, where a bootstrap pro- cedure provides confidence intervals for the HI. Compared with four model-based methods and a conventional LSTM, the non-invasive approach improves damage estimation performance, espe- cially under small-sample conditions. Mei et al. (2024) employ a variational Autoen- coder (AE) to model degradation of the oper- ating voltage signal of electromagnetic relays and perform time-dependent reliability (failure- probability) assessment. A high-fidelity physics- of-failure FE simulation of the relayâs electromag- netic and elastic subsystems is used to generate synthetic degradation trajectories, which are com- bined with a small set of experimental trajectories to train the variational AE. This then functions as a generative degradation model: operating volt- age trajectories are sampled over time, and at each time point, the fraction of generated samples exceeding a voltage threshold is used to estimate the time-dependent failure probability of a batch of relays. From a data perspective, training on the combined simulation and experimental tra- jectories outperforms training on either source 22 alone. From a modeling perspective, the vari- ational AE achieves lower reliability-assessment error and computational cost than both Gaussian Process (GP)- and LSTM-based variants. 4.3.4 Prognosis Five studies employing observational bias for prognosis have been identified, which are listed in Table 8. Y. Deng et al. (2023) leverage a 5-DOF dynamic bearing model for RUL pre- diction across different machines. The dynamic model describes vibration under crack propaga- tion and spall growth, with the associated dam- age parameters inferred via PF-based calibration. Both calibrated simulated data and real measure- ments are fed into a Bayesian N, which processes one-dimensional time-series features through a Gated Recurrent Unit (GRU) branch and two- dimensional time-frequency features through a CNN branch, jointly predicting the RUL. Within an adversarial TL setup, multiple domain dis- criminators are weighted according to the simi- larity of calibrated physical parameters, so that source bearings with degradation dynamics closer to the target bearing have a stronger influence during adaptation. Experiments on two bench- mark bearing datasets demonstrate more accurate RUL predictions than purely data-driven TL base- lines. Similarly, Y. Zhang, Feng, et al. (2023) generate full life-cycle vibration data using a 5-DOF dynamic bearing model. The resulting sim- ulated data serve as the source domain for a multilayer Transformer-based network with maxi- mum mean discrepancy-based domain alignment, which is trained to transfer degradation knowl- edge to measured vibration signals. Experimental results show that the proposed model substan- tially reduces RUL prediction errors compared to several state-of-the-art methods, particularly when only limited measured run-to-failure data are available. Y. Zhu, Cheng, et al. (2024) tackle RUL prediction for rolling bearings, proposing a framework that incorporates physics in three ways. In terms of observational bias, a physics- based degradation model is used to simulate full life-cycle vibration data. The simulated signals are subsequently fused with sensed data for fea- ture extraction, thereby enriching the training set used to train the BiLSTM. In combination with the other two mechanisms (see Sec. 4.5.4 and Sec. 4.6.4, respectively), the proposed frame- work outperforms purely data-driven baselines by achieving lower prediction errors. However, due to the absence of ablation studies, it is not pos- sible to isolate how much of the improvement is attributable to this particular strategy. Cutting-tool RUL prediction under scarce degradation data is studied by N. Zhang et al. (2024), who propose combining a physics-based cutting model with an improved inverse GP. An FE cutting model is used offline to gener- ate full-life wear trajectories, which are used for initial parameter estimation of the inverse GP. During operation, its parameters are updated online with measured wear via Bayesian inference. Experiments on three tool-wear datasets indi- cate improved predictive accuracy over a purely data-driven baseline. In their study, Herv Ěe de Beaulieu et al. (2024) focus on predicting the RUL of industrial systems without relying on labeled run-to-failure data. Linear closed-loop models of an aircraft cockpit temperature control system are identified from flight data and coupled with an exponential valve- stiction degradation model to generate nominal and degraded time series. An AE is trained only on nominal data to derive an unsupervised HI from reconstruction error, which is then forecast using an LSTM. The approach is validated through a reliability-based assessment using Weibull and Kolmogorov-Smirnov tests, showing internal con- sistency of predictions. However, the study does not report experimental comparisons to alterna- tive methods. 4.4 Inductive Bias Inductive bias is characterized by tailored inter- ventions to the model design. The identified stud- ies are organized by PHM task in Tables 9â12. A representative example is illustrated in Figure 6. 4.4.1 Fault Detection Two studies have been identified that employ inductive bias to address fault detection (see Tab. 9). S. Liu et al. (2025) address anomaly detection in large-scale multi-component systems with complex interdependencies. The authors pro- pose a model based on an AE and graphs that learns unit-level patch embeddings and models global relationships in multivariate time series in 23 Table 8All studies employingobservational biasto addressprognosis, listed in alphabetical order. Observational Bias for Prognosis ReferencePrior Physical KnowledgeUse Case TypeRepresentation Herv Ěe de Beaulieu, Jha, Garnier, and Cerbah (2024) Stiction model combined with a generic exponential degradation model Nonlinear difference equation with a time- varying parameter prescribed by an expo- nential function Air distribution system Y. Deng, Du, Wang, Shao, and Huang (2023) 5-DOF dynamic bearing model ODE systemBearing Y. Zhang, Feng, et al. (2023) 5-DOF dynamic bearing model ODE systemBearing N. Zhang et al. (2024)Cutting model (accounting for geometry, material, friction, and wear) FE-based numerical simulation Milling Y. Zhu, Cheng, et al. (2024) Vibration degradation modelAnalytic model defined by algebraic equations Bearing Table 9All studies employinginductive biasto addressfault detection, listed in alphabetical order. Inductive Bias for Fault Detection ReferencePrior Physical KnowledgeUse Case TypeRepresentation Y. Feng, Chen, Liu, Lv, and Wang (2022) Topological prior (sensor infor- mation regarding hardware layout or shared subsystems) Relation matrix fused with adjacency matrix Liquid rocket engine S. Liu, Chen, Liu, Wang, and Wang (2025) Topological prior (sensor infor- mation) Adjacency matrixLiquid rocket engine, train transmission sys- tem an unsupervised manner. Prior knowledge of sys- tem structure is encoded as a graph and incorpo- rated via an adjacency matrix, guiding the model to produce representations consistent with known system topology for both anomaly detection and component-level localization. Experiments on liq- uid rocket engine and train transmission datasets demonstrate improved accuracy and reliability, at a higher computational cost. In a closely related study also considering the liquid rocket engine sys- tem, Y. Feng et al. (2022) encode prior knowledge into the adjacency matrix as well. Topological relationships of physical sensors in terms of hard- ware layout or shared subsystems are leveraged. Improvements in both accuracy and generalizabil- ity are achieved, compared to strong non-graph- based and graph-based baselines. 4.4.2 Diagnosis As shown in Table 10, three studies employ induc- tive bias to target diagnosis. Zeng et al. (2025) tackle rolling bearing fault diagnosis by embed- ding bearing fault frequency information into a CNN via a novel Frequency-Aware Convolution (FAC) layer (see Fig. 6). Vibration signals are first transformed into time-frequency maps via Contin- uous Wavelet Transform (CWT), which serve as inputs to the network. The architecture is built from multiple stacked units, each comprising a residual block, the FAC layer, and an Efficient Channel Attention (ECA) module (Q. Wang et al., 2020). Using the characteristic fault frequen- cies of the inner ring (BPFI), outer ring (BPFO), and rolling body (BSF), the FAC layer generates a set of sinusoidal waveforms to form frequency 24 Application of Frequency Aware Mechanism-Based Physical Information Convolutional Neural Network in Rolling Bearing Fault Diagnosis Zeng et al., 2025 Use Case:Bearing fault diagnosis Prior Knowledge:Fault characteristic frequencies â˘Reliable diagnostic features: â˘BPFI = 2 1 + â˘BPFO = 2 1 Incorporation:Frequency-aware convolutional layer â˘Following a bandpass filter design, the frequency response kernel encodes the bearingâs fault frequencies as corresponding sinusoidal waveforms â˘Multiplication with a standard kernel yields enhanced kernels Input LayerLayer Output Residual Block ECA FAC Layer BPFI BPFO BSF â= 1 Frequency-aware Convolution 3 2 Fig. 6A representative example of inductive-bias approaches is proposed by Zeng et al. (2025), demonstrating how prior physical knowledge can be introduced into the model design. Own illustration based on the corresponding study. See the original work for full technical details. Table 10All studies employinginductive biasto addressdiagnosis, listed in alphabetical order. Inductive Bias for Diagnosis ReferencePrior Physical KnowledgeUse Case TypeRepresentation C. Gao, Wang, Guo, Wang, and Yi (2024) Vibration model for a local- ized single-point defect, bearing fault characteristic frequencies Analytical time-domain signal model, algebraic equations Bearing Jin et al. (2024)Structural topology of the wind turbine and qualitative causal relationships Directed graph over selected sensor signals (adjacency matrix) Wind turbine Zeng et al. (2025)Fault characteristic frequenciesAlgebraic equationsBearing response kernels (one per frequency) with a band- pass profile. A learnable shift parameter, initial- ized to zero and constrained within a physically meaningful range, allows these kernels to dynami- cally adapt to varying vibration signals across dif- ferent operating conditions while remaining close to the analytical fault frequencies. Each frequency response kernel is multiplied element-wise with a standard convolutional kernel to form enhanced kernels that selectively emphasize the correspond- ing fault-related frequency bands while suppress- ing irrelevant components. The proposed model demonstrates superior accuracy, noise robustness, and generalization across varying loads compared with conventional CNN-based models, as vali- dated on two datasets. Additionally, feature-space visualization (based on t-distributed stochastic neighbor embedding) reveals physically consistent clusters aligned with bearing fault mechanisms. A conceptually similar, yet structurally different approach is proposed by C. Gao et al. (2024), who use a set of subnetworks to target small- sample learning. Each subnetwork is tailored to a specific failure mode (class). Based on the cor- responding bearing fault characteristic frequency, a kernel that simulates an ideal bearing fault vibration replaces the data-driven kernel of the first convolutional layer of an otherwise conven- tional CNN, while the healthy-condition subnet- work retains a standard data-driven first layer. These subnetworks are trained independently as binary classifiers to specialize in fault-specific fea- ture extraction from vibration signals. Fusing the extracted features and feeding them to a unifying 25 classifier enables multi-class diagnosis of rolling bearings. Validated on two datasets, the proposed model improves small-sample accuracy compared to both classical baselines and CNN-based models. In their work, Jin et al. (2024) propose a spatio-temporal GNN for wind turbine condi- tion monitoring. They construct a directed graph based on prior knowledge of turbine structure and inter-component causal relationships, with nodes representing sensor signals and edges encod- ing structural and causal connections. On this prior graph, a graph attention network layer cap- tures spatial dependencies between signals, while global-local attention and recurrent layers model their temporal evolution under healthy operation. The network is trained to predict the next-step values of all signals, and deviations between pre- dicted and actual values are monitored at both the graph and node levels. By doing node-level anomaly detection, it does perform component- level fault localization, which is a practically meaningful form of diagnosis. A multi-node fault propagation chain, constrained by the prior graph, is used to distinguish true faults from scattered false alarms and to identify the origin of abnormal behavior. Case studies on a real turbine with a generator bearing failure and several false alarms show comparable early warning capability, fewer false alarms, and more interpretable monitoring compared with purely data-driven DL baselines. 4.4.3 Health Assessment Inductive bias is used in 17 studies on health assessment (see Tab. 11). To estimate the SOH of lithium-ion batteries, H. Huang et al. (2022) design a custom kernel for GPR based on an empirical degradation model that captures both linear and exponential decay over cycling. The kernel hyperparameters are initialized using the parameters of the empirical model (identified via least squares), facilitating subsequent optimiza- tion. Benchmarking against standard Gaussian and Mat Ěern kernels shows that the proposed ker- nel consistently achieves lower prediction errors on two experimental datasets. Fu et al. (2024) pro- pose a Recurrent Neural Network (RNN) whose architecture is derived from a discretized second- order ECM, in which the open-circuit voltage is modeled by an MLP. Treating the ECM parameters as trainable variables enables accu- rate online parameter identification and terminal- voltage prediction with low computational com- plexity. Experimental results demonstrate that the proposed method outperforms both a white-box neural circuit model and an enhanced ECM. Fur- ther analysis reveals a linear relationship between the identified ohmic resistance and remaining capacity, enabling accurate SOH estimation. Also addressing SOH estimation of lithium-ion batter- ies, Lehmann et al. (2024) propose a stress-factor- based aging model that is parameterized using both laboratory aging tests and data from an elec- tric bus fleet. The model consists of cyclic and cal- endric stress maps and a square-root aging curve. The stress maps are represented by NNs that share this fixed aging-curve structure with a separately fitted empirical model. Capacities predicted by the empirical model at collocation points are included in the training objective, which ties the learned stress maps to the established functional relation- ships in regions that are sparsely covered by data. Using TL to adapt the aging-curve parameters to different cell types and to the fleet, the approach improves capacity and SOH estimates at check-up tests compared with an equal-stress baseline and with the coupled model without adaptation. For lithium-ion battery health management, L. Qin et al. (2025) propose a predictor-estimator frame- work that jointly handles SOC, internal resis- tance, and maximum available capacity. A linear- exponential two-stage degradation model with recursive least-squares updating and Bayesian knee-point detection predicts the evolution of sev- eral health indicators (SOH, a resistance-based HI, efficiency, and SOC at discharge start), from which future capacity and resistance trajectories are derived. Three lightweight neural estimators then infer SOC, resistance, and capacity from operational time series, where physics is embedded by hard-wiring Coulomb counting dynamics with learnable efficiency and initial-SOC corrections for SOC estimation. The second estimator couples an MLP with an equivalent-circuit Ordinary Differ- ential Equation (ODE) model whose parameters are learned under physical bounds for resistance estimation, while the capacity estimator exploits these latent variables via channel-attention and temporal convolutions. Experiments on two public battery datasets show that this framework pro- vides more accurate degradation prediction and 26 Table 11All studies employinginductive biasto addresshealth assessment, listed in alphabetical order. Inductive Bias for Health Assessment ReferencePrior Physical KnowledgeUse Case TypeRepresentation Bajarunas, Baptista, Goebel, and Chao (2024) Causal relationship (sensors, operating conditions, degrada- tion) Architectural con- straint (and corre- sponding regularization terms) Turbofan engine, lithium-ion bat- tery Cheng, Zhang, Wang, Yang, and Li (2024) Thermodynamic and structural relations of the gas turbine Adjacency matrixGas turbine Ellis et al. (2022)Model relating crack length and first natural frequency Non-stationary GP prior Turbomachine rotor blade Fu et al. (2024)Second-order ECMParametric state-space model Lithium-ion bat- tery Hao et al. (2023)Monotonicity assumptionActivation functionMilling H. Huang et al. (2022)Empirical degradation modelAlgebraic equationLithium-ion bat- tery Lehmann, Berendes, Kratzing, and Sethia (2024) Cyclic stress model, calendric stress model and aging curve model Algebraic equationsLithium-ion bat- tery S. Li, Li, and Zhu (2025) Monotonicity assumptionArchitectural con- straints Milling Z. Ma, Zhao, Dai, and Chen (2023) Degradation dynamics (Wiener process) Stochastic discrete-time state-space model Milling L. Qin, Sun, Sun, and Xia (2025) First-order ECM with hys- teresis and Coulomb counting model ODE-based layersLithium-ion bat- tery F. Xie et al. (2024)Monotonically increasing one- dimensional degradation state in the range [0,1] Architectural con- straint (with associated loss terms) Semiconductor (insulated gate bipolar transistor) Yucesan and Viana (2019, 2020, 2021, 2022, 2023) Cumulative fatigue damage model with lubricant influence First-order ODE, alge- braic equations Bearing K. Zhu, Huang, Li, and Lin (2023) Empirical tool wear modelsAlgebraic equationsMilling state estimation than state-of-the-art baselines, generalizes well across different chemistries, tem- peratures, and loading conditions, and achieves these benefits with very compact estimator net- works and low computational overhead. Yucesan and Viana (2019) propose a frame- work for estimating wind-turbine main bear- ing fatigue life by combining a physics-based cumulative-damage model for bearing fatigue with a data-driven model for grease degrada- tion, organized within a recurrent model that models damage accumulation over time. Bear- ing fatigue is determined incrementally at each timestep by computing the fatigue damage rate from the current load and speed using an L10- based bearing life relation and accumulating dam- age via Palmgren-Minerâs rule. Simultaneously, an MLP predicts increments in grease degradationâ a hidden variable affecting fatigue calculations through viscosity and contamination factorsâ but is only indirectly supervised through peri- odic grease observations. By embedding these components within a custom recurrent cell, the model captures both the well-understood physics of bearing fatigue and the complex dynamics of grease degradation that are difficult to model from first principles. While promising, the results provide no quantitative performance metrics and 27 lack baseline comparisons, limiting the ability to fully assess the approachâs effectiveness. Even though the model is elaborated upon in sub- sequent works (Yucesan & Viana, 2020, 2021, 2022, 2023), the fundamental approach remains unchanged from a PIML perspective. Studying tool wear estimation in high-speed milling, S. Li et al. (2025) propose embedding an architectural constraint into a GRU network. To account for the irreversible nature of tool wear, monotonicity is enforced via constrained hidden- state updates and using Rectified Linear Unit (ReLU) activations. This ensures that predicted wear values cannot decrease over time, which improves both physical plausibility and predictive accuracy compared to unconstrained data-driven models. See Section 4.5.3 for a description of the learning bias incorporated by S. Li et al. (2025). A similar approach is adopted by Hao et al. (2023), where a softplus activation function is incorporated into the network architecture prior to the output layer. This design choice explicitly accounts for the monotonic degradation behav- ior of milling tool wear, ensuring nondecreasing wear predictions over time. As a result, the pro- posed model likewise achieves improved predictive accuracy compared to purely data-driven base- line approaches. K. Zhu et al. (2023) propose a GPR-based approach for predicting tool wear. Prior knowledge is incorporated through three physics-based tool wear models (generalized Tay- lor formula, a cubic polynomial wear-time law, and a generic flank wear model) that describe the evolution of tool flank wear over cutting time. By adopting these wear laws as the prior mean function, the model supports small-sample training, continual online updating, and substan- tially improved extrapolation. Comparative exper- iments confirm that the proposed approach sig- nificantly reduces prediction error and improves robustness relative to both standalone physics- based models and purely data-driven approaches. Z. Ma, Zhao, et al. (2023) also address tool wear monitoring in milling. A physics-based state-space model based on a Wiener process-inspired wear law serves as prior knowledge within a GP-based probabilistic state-space model. A PF estimates the unknown posterior distribution of the degra- dation state. Compared with data-driven base- lines such as CNN, LSTM, and Support Vector Regression (SVR), as well as purely physics-based approaches, the method yields a better predictive performance and tighter confidence intervals for tool replacement decisions. The method is also suitable for prognosis tasks and RUL prediction. Bajarunas et al. (2024) aim to leverage gen- eral knowledge about degradation to broaden the applicability of unsupervised HI estimation across various systems. They use assumed causal relationships between sensor readings, operating conditions, and the degradation to design the architecture of a convolutional AE: the encoder maps sensor readings to a scalar latent variable, while the decoder reconstructs sensor readings from this latent variable and operating condi- tions, thereby forcing the latent variable to encode degradation information. To further shape this latent variable into a suitable HI, they comple- ment this inductive-bias approach with loss terms that guide HI properties desired in PHM, such as monotonicity and trendability, as well as an optional term that encourages consistency with degradation trends derived from reliability the- ory. The approach is validated on both turbofan engine data and lithium-ion battery data, lead- ing to improvements in prediction performance and out-of-distribution robustness compared to residual-based baselines. With a subsequent CNN, the approach is extended for RUL prediction, yet the PIML aspects in this paper solely correspond to health assessment. To assess the degradation state of insulated gate bipolar transistor modules under varying operating conditions, F. Xie et al. (2024) pro- pose an LSTM-based AE designed to disentan- gle degradation from operating conditions. The encoder compresses inputs into a low-dimensional latent vector, reserving a single scalar for the degradation index while the remaining dimen- sions capture variations in operating conditions. Two decoders are then employed: one reconstructs non-degraded behavior from the condition-related latents alone, and the other reconstructs degraded behavior using the full latent vector. This design forces degradation information to flow through a single neuron and yields an interpretable HI suitable for online monitoring. Complementing this inductive-bias approach, the authors employ additional loss terms that guide this single health- indicator neuron to satisfy both monotonicity and range constraints. This leads to better results in 28 terms of prediction performance and physical con- sistency compared to the inductive-bias approach alone. Cheng et al. (2024) assess gas turbine health via a spatio-temporal GNN. Thermodynamic and structural knowledge is encoded by constructing a temporal graph over key monitored parameters, whose edges combine kNN-based data correla- tions with links derived from small-deviation com- pressor, combustor, and turbine equations. This topology effectively constrains how information propagates between variables, enabling accurate health assessment by mapping multivariate time series to discrete health stages, as corroborated by corresponding ablation studies. As part of a broader framework, Ellis et al. (2022) address the health assessment of turboma- chine rotor blades by estimating root crack length from blade tip timing-derived natural frequen- cies under scarce inspection data. A physics-based model (FE simulations with an unscented trans- form) is first built to map natural frequency to crack length, and this ensemble is then used as the prior mean and covariance of a GPR model. The latter is conditioned on crack-length measurements obtained via non-destructive test- ing during routine maintenance. The proposed model preserves physically plausible behavior in data-sparse regions while correcting system- atic errors near observed non-destructive testing points, and it consistently outperforms both the pure physics-based model and several purely data- driven regressors. 4.4.4 Prognosis Table 12 provides an overview of the 15 identi- fied studies regarding inductive bias for prognosis. Nascimento et al. (2021) propose a method for lithium-ion batteries that embeds core electro- chemical relations to predict voltage discharge curves (and thereby end-of-discharge time) under varying loads and to forecast aging-induced capac- ity fade and resistance growth. A reduced-order model based on the Nernst and Butler-Volmer equations is implemented as a recurrent cell, while MLPs replace the non-ideal voltage (activity) terms that are difficult to capture analytically. The framework treats lumped internal resistance and maximum available charge as cell- and age- dependent parameters, and models their evolution with cumulative discharged energy using varia- tional ensemble learning to obtain quantitative aging indicators and uncertainty-aware forecasts. Despite extensive experiments, no quantitative results are reported that demonstrate improve- ments over purely data-driven baselines. Method- ologically, this approach is closely related to the work of Yucesan and Viana (2019) and its follow- up studies (see Sec. 4.4.3). Bai et al. (2023) address lithium-ion battery capacity prognostics using a two-stage framework. In the first stage, an N combined with a dual Extended Kalman Filter (EKF) is employed for online estimation of the SOC and capacity from voltage and current measurements, producing capacity trajectories. In the second stage, these trajectories are used as inputs to a GPR-based degradation model to forecast future capacity evolution. Two inequal- ity constraints are imposed on the GPR, ensuring that capacity predictions remain bounded and monotonically decreasing with respect to the cycle number. The constrained GPR outperforms its unconstrained version, as well as three additional baseline methods in terms of both predictive accuracy and reduced uncertainty. Addressing RUL prediction for rolling bear- ings, C. Yin et al. (2025) focus on physically consistent modeling of degradation. The paper proposes a method that leverages phase space reconstruction to transform vibration signals into trajectories, turning RUL prediction into a vari- ation estimation problem. Prior knowledge about the monotonic nature of bearing degradation is integrated into a 1D-CNNâs final activation func- tion, ensuring that the predicted RUL cannot increase unrealistically. This improves robustness, smoothness, and physical plausibility of predic- tions compared to purely data-driven models. Comparative experiments under varying work- ing conditions demonstrate that the approach outperforms state-of-the-art methods in both pre- dictive accuracy and generalizability. Nguyen et al. (2023) address the limitations of purely data- driven GANs for RUL prediction, including insta- bility, sample inefficiency, and lack of physical consistency. Their proposed architecture attaches a differentiable fuzzy logic module to the out- put of a conditional GANâs generator. Within this module, the standard product aggregation opera- tor is replaced by a dataset-specific physics model 29 Table 12All studies employinginductive biasto addressprognosis, listed in alphabetical order. Inductive Bias for Prognosis ReferencePrior Physical KnowledgeUse Case TypeRepresentation Abiria, Wang, Zhang, Liu, and Jin (2025) Basquinâs law, nonnegativity assumption Algebraic equations, differential equations, architectural con- straints Additive manufac- turing Badora et al. (2023)Parisâ lawDifferential equationHigh-pressure nozzle of an industrial gas tur- bine Bai, Su, Rahman, and Wang (2023) Bounded and monotonically decreasing capacity fade over cycles Inequality constraintsLithium-ion bat- tery X. Cai, Zhang, Yu, and Xie (2025) System/sensor topologyKnowledge graphTurbofan engine, milling Dourado and Viana (2019, 2022) Walker model for fatigue crack propagation Recurrent algebraic equation Aircraft fuselage panels C. Jiang, Zhong, Choi, and Youn (2025) Parisâ law for fatigue crack growth ODE discretized to a damage accumulation model Aluminum speci- mens Nascimento, Corbetta, Kulkarni, and Viana (2021) Nernst and Butler-Volmer equations Algebraic equationsLithium-ion bat- tery Nguyen, Singh, and Rai (2023) Spall-growth model, modified Eyring model Algebraic equationBearing, turbofan engine Qiang, Shi, Liu, Ren, and Shi (2023) Empirical linear tool wear model Algebraic equationMilling L. Qin, Zhang, Sun, and Zhao (2024) Power equations of low- and high-pressure compressor, high- speed shaft dynamics Algebraic equation, ODE Turbofan engine C. Yin, Li, Wang, and Dong (2025) Monotonicity assumptionArchitectural con- straints Bearing Y. Zhang, Wang, Zhang, Dui, and Chen (2025) Diamond-shaped wear particle model, Archard wear model and wear model by Zou, Huang, and When (1996) Algebraic equationsAxial piston pump Z. Zhou et al. (2023)Nonnegativity assumptionArchitectural con- straint Turbofan engine, bearing M. Zhou, Li, Cao, Ma, and Xu (2025) Thermal cycle modelFixed sensor associa- tion graph (adjacency matrix) Turbofan engine (a spall-growth model for bearings; a modified Eyring model for turbofan engines). The generator thus learns fuzzy implications whose values serve as parameters of the physics model, constrain- ing predictions to physically realistic solutions while the adversarial training signal still flows end-to-end through both the fuzzy and physics layers. Experiments on two different datasets (bearings and turbofan engines) show reduced pre- diction errors. Additional experiments, in which the dataset size is iteratively reduced, demon- strate improved data efficiency compared to a 30 conventional LSTM. Z. Zhou et al. (2023) tar- get RUL prediction as a time-varying trajectory modeling problem rather than a point-wise esti- mation. Building on Neural ODE (R.T.Q. Chen, Rubanova, Bettencourt, & Duvenaud, 2018), the proposed approach integrates additional prior physical knowledge about smooth degradation behavior. Physically meaningful RUL trends are enforced by incorporating a nonnegative bounded function prior to providing the final prediction. Furthermore, a dynamic learning scheme utilizing a super-network (Y. Wu, Liu, Huang, Zhang, & Van Gool, 2021) and deep RL enables adaptive time-dependent network architectures, improving the modelâs ability to capture underlying degra- dation dynamics. This approach results in more stable and interpretable RUL predictions that align with real-world degradation processes. The proposed method shows smooth and accurate pre- diction results compared to common data-driven methods such as Residual Network (ResNet) and BiLSTM, as demonstrated through experiments on both bearing and turbofan engine datasets. Qiang et al. (2023) study tool wear predic- tion in milling under varying cutting parameters, where new operating conditions offer only lim- ited labeled wear data. To tackle this challenge, an instance-based regression transfer algorithm (Two-stage TrAdaBoost.R2, proposed by Pardoe and Stone (2010)) is combined with a recurrent GPR base learner. The GPRâs mean function encodes empirical physical relations between flank wear, cutting power, and previous wear to cap- ture time-accumulation and degradation trends, while the kernel models residual nonlinearities. Using only about 30 % of early-life wear data, the framework accurately extrapolates the full wear trajectory and yields tight confidence inter- vals. The proposed framework is evaluated against three alternative approaches: an otherwise identi- cal framework employing a standard (uninformed) GPR, a recurrent GPR, and an LSTM, with the latter two not incorporating TL. Experiments show that the proposed framework achieves sub- stantially lower prediction errors, better tracking of late-life wear growth, and more stable perfor- mance across different cutting-parameter combi- nations. Unlike many studies that rely only on temporal sensor sequences, X. Cai et al. (2025) take spatial interactions among multiple sensors into account. Two use cases (milling and turbofan engines) are considered to leverage system topology and sensor placement information. In a first step, embed- dings are learned that represent the real-world topological structure. This is done by employing an energy-based knowledge embedding algorithm. Distances between embeddings are used to con- struct graph edges and an initial weighted adja- cency matrix, whose weights are then dynamically updated via an attention mechanism. These are fed into a model consisting of spatial modules (graph convolutional network and attention mech- anism) as well as temporal modules (LSTM) and a final fully connected layer for direct RUL predic- tion. In ablation scenarios considering the turbo- fan use case, it is shown that temporal modules, spatial modules, and the knowledge encoding in the adjacency matrix contribute to the model per- formance. In comparison with multiple baselines (e.g., CNN, LSTM, and Bayesian models), the proposed method achieves superior performance in direct RUL prediction tasks in the majority of scenarios. Comparable results are also reported for the milling use case. L. Qin et al. (2024) address the inadequate interpretability prevalent in DL methods applied to RUL estimation, with a par- ticular focus on turbofan engines. The proposed model comprises an augmenter for noise filtering, interpolation, and unobservable state estimation, and an estimator for RUL prediction. The aug- menter is based on a Neural ODE framework that integrates physical models, data-driven mod- els, and a Runge-Kutta ODE solver, all trained end-to-end. The estimator combines LSTM-based encoding with feature and temporal attention. As prior knowledge, power equations of the low- pressure compressor and high-pressure compres- sor, shaft-speed dynamics, and stall-margin and efficiency-modifier equations are embedded into the augmenter. Compared to a variety of data- driven baselines, the proposed method proves to be superior in terms of predictive performance. Also targeting the RUL prediction of turbofan engines, M. Zhou et al. (2025) embed thermody- namic prior knowledge into a spatio-temporal N via graph construction. In the spatial branch, ther- modynamic cycle and engine-structure knowledge are used to build an association graph between sensors, which is fused with a gray-relation graph to form a fixed adjacency matrix for a multi- layer graph attention network with pooling. In 31 the temporal branch, an LSTM extracts temporal features, while a temporal-pattern attention mod- ule derives time-invariant features from its hidden states, which together form the temporal-domain features. An attention module then fuses temporal and spatial features for RUL prediction. Exper- iments on C-MAPSS show that the proposed model achieves consistently lower prediction errors than standard DL and other spatio-temporal base- lines. Y. Zhang, Wang, et al. (2025) address the degradation of hydraulic piston pumps, which they characterize as arising from the interplay of several factors, most notably the progressive wear of internal friction pairs together with fluctuating external load and operating conditions. To capture this behavior, they propose a framework for pre- dicting the RUL of hydraulic piston pumps built around an LSTM. Wear laws for three key friction pairs (valve plate, piston, slipper) are embedded in an end-to-end design, with wear model param- eters updated jointly with the LSTM weights. These adaptive wear models convert monitoring data into degradation indicators, which the LSTM uses to predict return oil flow. RUL is estimated as the time until the predicted flow exceeds a critical threshold. The proposed method outper- forms conventional baselines (including SVR, RF, and LSTM), though more advanced prognostics methods were not evaluated. Aging aircraft fleets are studied in the works ofDourado and Viana (2019, 2022), with a specific focus on fatigue crack propagation in aircraft fuselage panels. In both contributions, a physical fatigue crack propagation model (the Walker model, an adapted version of the well- known Paris law) is embedded directly into a custom recurrent cell, while a data-driven branch within the cell learns a correction term that accounts for corrosion effects not captured by the Walker model. However, the authors do not bench- mark their method against purely data-driven or purely physics-based baselines, making it diffi- cult to quantitatively assess the added value of the proposed approach. Similar to Nascimento et al. (2021), this approach is also methodologically related to the work of Yucesan and Viana (2019) and its follow-up studies. C. Jiang et al. (2025) study probabilistic prognosis of fatigue crack growth in metallic specimens. Monte Carlo simu- lations of a physics-based model (Parisâ law) are truncated using a standard GP fitted to current observations, whereas the resulting trajectories are used to construct a non-stationary prior mean and covariance for the final GP. Experiments show that incorporating these priors markedly improves extrapolation performance and predic- tive accuracy compared to both a standard GP and a PF. Badora et al. (2023) introduce a custom RNN cell designed to model fatigue crack growth in a gas turbine nozzle. An N estimates the stress intensity factor range at shutdown, while a physics-based part then applies Parisâ law to com- pute the crack length increment due to fatigue. The model accurately predicts crack growth over multiple cycles, even with limited observed data, and outperforms standard regression models in terms of predictive performance. In their work, Abiria et al. (2025) tackle the challenge of predict- ing fatigue life in additively manufactured alloys, where cyclic loading leads to microscopic damage accumulation and eventual failure. The authors address this problem by integrating prior knowl- edge into a conventional N through modified activation functions derived from Basquinâs law, a modified Paris law, and a nonnegativity condition. The proposed model shows better generalization capabilities compared to other physics-informed and purely physics-based variants. 4.5 Learning Bias Learning bias is characterized by introducing prior physical knowledge into the learning algorithm, frequently realized via a composite loss function. Such a composite loss typically combines a data- fidelity term, which minimizes the discrepancy between predictions and observations, with one or more physics-informed terms that penalize viola- tions of the governing equations, boundary con- ditions, or other physical constraints, each scaled by a weighting coefficient that controls its relative influence on training. The identified studies are organized by PHM task in Tables 13â16. A repre- sentative example is illustrated in Figure 7, which includes a composite loss function. 4.5.1 Fault Detection Table 13 lists the three studies that employ learning bias for fault detection. X. Xu and Liu 32 Table 13All studies employinglearning biasto addressfault detection, listed in alphabetical order. Learning Bias for Fault Detection ReferencePrior Physical KnowledgeUse Case TypeRepresentation A. Wang, Qin, Yuan, Zhao, and Sun (2025) Wave-type vibration model of spline shaft Nonhomogeneous 1D wave PDE Aero-engine involute spline coupling S. Wang et al. (2024)Multi-energy model of a robot joint, multi-joint rigid-body dynamics Algebraic energy- balance equations and recursive Newton-Euler dynamic equations Industrial multi- axis robot X. Xu and Liu (2024)Damage index model, mono- tonic stress-strain relation Algebraic equationsCarbon fiber rein- forced polymer laminates (2024) target Lamb-wave-based fatigue damage detection in carbon fiber reinforced polymer lam- inates by augmenting a CNN with an additional branch that estimates global stiffness degrada- tion from time-frequency images of guided-wave signals obtained via CWT. Their approach incor- porates additional loss terms based on a damage index model and a stress-strain constraint, thereby regularizing training toward physically consistent progressive degradation. The damage index model relates stiffness degradation and off-axis angle to normalized power spectral density changes of Lamb-wave responses, converting the networkâs predicted stiffness into pseudo-damage labels. The stress-strain constraint enforces monotoni- cally increasing strain under constant load, penal- izing non-monotonic strain trajectories derived from the predicted stiffness. Using only data from one carbon fiber reinforced polymer layup, the method generalizes to unseen layups with sub- stantially improved cross-structure detection per- formance compared to a standard CNN, and the resulting path-level damage predictions support accurate delamination localization. A. Wang et al. (2025) propose a soft sens- ing framework for estimating difficult-to-measure aero-engine variables. The approach extends PINNs to nonhomogeneous Partial Differential Equations (PDEs) with unknown, unmeasurable driving terms by training two coupled NNs in a hierarchical (alternating) optimization scheme: one approximates the PDE solution, the other the unknown source term. By employing a joint loss in which the learned source is embedded in the PDE residual, the solution is regularized toward PDE- consistent behavior, while the source is simultane- ously constrained to produce driving terms that are compatible with both the measurements and the governing equation. A recurrent-prediction term further refines the solution using delayed hard-sensor and soft-sensor outputs to mitigate information loss due to sparse sampling and unmeasured sources. Applied as a virtual vibra- tion sensor on an aero-engine spline-coupling test rig, the method achieves substantially lower pre- diction errors than standard PINNs and supports a proof-of-concept anomaly detection example for spline-coupling health monitoring. By using a convolutional AE, S. Wang et al. (2024) estimate joint electrical current from multivariate sensor data (e.g., motion, temper- ature, and vibration) for anomaly detection in industrial robot systems. Physical knowledge from energy conservation in the joints (multi-energy model) and Newton-Euler multi-joint dynamics is incorporated via additional loss terms, enforc- ing energy-flow consistency and torque coupling between joints. Using the Kullback-Leibler diver- gence between estimated and measured current as a health indicator, the proposed method detects injected motor and reducer faults on real factory robots with superior accuracy, outperforming sev- eral state-of-the-art time-series anomaly detection methods. 4.5.2 Diagnosis Nine studies employing learning bias for diagnosis have been identified, as summarized in Table 14. Qiao et al. (2024) focus on the small-sample 33 Table 14All studies employinglearning biasto addressdiagnosis, listed in alphabetical order. Learning Bias for Diagnosis ReferencePrior Physical KnowledgeUse Case TypeRepresentation Chao, Hu, and Liu (2025) Discharge-pressure and internal leakage model, volumetric effi- ciency definition Nonlinear ODE, alge- braic equation Axial piston pump C. Dong et al. (2025)Mass and momentum conser- vation in fluid dynamics, valve and periodic boundary condi- tions PDEs, algebraic equations Axial piston pump Y. Huang, Tang, Yang, and Ming (2025) Structural causal model for gearbox vibration data Algebraic equationPlanetary gear- box, wind turbine gearbox R. Li et al. (2024)Robot dynamic modelSecond-order ODEIndustrial robot Qiao, Liu, Huang, and Wu (2024) Bearing fault frequenciesNumerical valuesBearing S. Sun, Peng, Zhou, Zhang, and Wang (2024) 4-DOF multi-body bearing dynamics model Second-order ODE sys- tem Bearing Tang et al. (2024)Demand for consistent time and frequency domain latent representations Invariance lossBearing, gearbox Z. Xu, Zhao, Wang, and Bashir (2024) Bearing fault frequenciesAlgebraic equationsBearing Y. Zhu, Zi, Li, and Xu (2024) Bearing dynamic model, domain-invariant features per fault case Second-order ODE, progressive consistency causal factorization loss Bearing problem in bearing fault diagnosis under variable operating conditions. A 1D-CNN with a sequen- tial temporal attention module is trained within a contrastive learning framework to obtain dis- criminative fault representations from vibration signals. Prior knowledge enters as analytically derived fault characteristic frequencies, which are provided as additional targets and predicted from the learned embedding via an auxiliary fully con- nected head. The loss between predicted and known characteristic frequencies, weighted within a composite loss alongside cross-entropy and contrastive loss, softly enforces that the latent representation encodes these physically meaning- ful frequency features, thereby guiding the net- work toward fault-relevant structure in the data. The resulting model outperforms purely data- driven baselines on two publicly available bearing datasets, with ablation studies confirming the ben- efit of the embedded learning bias, particularly in small-sample scenarios. To address distribution shifts arising from variations in bearing structure and operating conditions, Y. Zhu, Zi, et al. (2024) propose a domain generalization method capable of extracting domain-invariant features for fault diagnosis. The model consists of a Fourier-based low-pass filtering module with learnable parame- ters, a CNN-based feature extractor and a fully connected classifier. Two kinds of regularization terms are incorporated: a dynamic embedding loss to guide features that account for the state of the target machineâs bearing dynamics and a progressive consistency causal factorization loss. The latter guides correlation for features of the same fault case across different machines and oper- ating conditions, while discouraging correlation between features of different fault cases. There- fore, domain-invariant features can be realized. Experiments show that the proposed approach demonstrates superior performance in terms of accuracy, generalizability, and interpretability in comparison to common domain generalization methods. Z. Xu et al. (2024) also incorporate fault characteristic frequencies of bearings into 34 their approach. They use a dual-branch AE to reconstruct bearing vibration data in the fre- quency domain for both real and imaginary parts of the spectrum. A modified loss function that makes use of masked target data is used for reconstruction. The mask highlights bands rele- vant to fault frequencies. Thus, a robust latent space that is biased toward fault frequencies rel- evant to fault cases is obtained as input for a subsequent classifier. Their approach outperforms state-of-the-art methods in terms of accuracy and well-separated feature spaces. To effectively tackle label-free fault diagnosis for rolling bearings, the proposed framework (S. Sun et al., 2024) combines a contrastive-learning backbone with a dynamics- embedding network based on sparse identification of nonlinear dynamics (Champion, Lusch, Kutz, & Brunton, 2019): a coordinate encoder reconstructs 4-DOF latent states from 1-DOF acceleration via delay embedding, and a physics-based equation library derived from a 4-DOF multi-body bear- ing model is used together with a sparse coeffi- cient matrix to infer the fault type. Physics-based constraints are imposed via the loss function, which enforces consistency between latent acceler- ations, reconstructed measurement signals and the accelerations predicted by the equation library, enabling the network to learn both the latent dynamics and a sparse, interpretable governing equation from raw signals. Experiments on both simulated and experimental bearing data show that the proposed framework can correctly dis- tinguish between inner-race, outer-race and roller faults without labels while providing physically meaningful diagnostic explanations. Tang et al. (2024) propose a self-supervised learning framework that addresses the small- sample problem in rotating machinery fault diag- nosis, studying both bearings and gearboxes. Two CNNs are used as feature extractors for the time domain and frequency domain, respectively. Dur- ing pretraining, an additional loss function incor- porating a distance metric is employed to obtain time-frequency domain invariant latent embed- dings. In the downstream task, the network is fine-tuned on the limited amount of labeled data to learn to diagnose the fault underlying the rotating part, showing superiority over alternative approaches both in terms of accuracy and data efficiency. Aiming to incorporate prior knowledge into domain generalization methods, Y. Huang et al. (2025) propose a causal learning network based on ResNet18. It is used to extract inde- pendent and causal features regarding the relation between the sensor data of gearboxes and distinct fault cases for diagnosis. Losses for an adver- sarial mask and an autocorrelation matrix, both founded on a structural causal model for vibration data, are incorporated to favor the aforemen- tioned properties. Compared to other domain gen- eralization methods, the approach demonstrates superior performance in terms of accuracy and class-distinguishability. The integration of model-based knowledge into diagnostic methods proves challenging in the field of axial piston pumps. Therefore, C. Dong et al. (2025) present a PINN framework serving as a high-frequency virtual dynamic flow meter in axial piston pumps by predicting pump flow ripple. The framework integrates fundamental physical principles of hydraulic systems, such as mass and momentum conservation, boundary condi- tions relevant to pump operation, and periodic characteristics of pump behavior into the loss function. The method facilitates robust pump fault diagnosis. Overall, the study validates the effectiveness of the proposed approach through numerical simulations, demonstrating close agree- ment with reference solutions, and through experi- mental investigations, showing that predicted flow ripples consistently reflect expected fault charac- teristics. Building on PINNs, Chao et al. (2025) tackle wear detection in axial piston pumps by reconstructing the discharge pressure while simul- taneously inferring the fluid film thicknesses at the pumpâs main friction pairs. Following the stan- dard PINN framework, an analytically derived ODE for the time derivative of discharge pres- sure is incorporated into the loss function, with internal leakage flows scaling cubically with film thickness. To stabilize the joint estimation of mul- tiple wear-related parameters with different scales, these parameters are learned as bounded variables via sigmoid-based range constraints. In a sequen- tial step, the identified film thicknesses are used in analytical formulas for volumetric efficiency and Cohenâsdeffect size, providing physically inter- pretable wear indicators and enabling localization of the worn friction pair. Experiments on a real pump with naturally worn components demon- strate accurate pressure reconstruction, physically 35 plausible thickness estimates, and correct iden- tification of the valve plate and cylinder block pair as the worn pair, although comparisons with alternative methods are not reported. Using available proprioceptive signals instead of external measurements, R. Li et al. (2024) propose an approach for fault diagnosis of indus- trial robots. The framework consists of an encoder based on GRU and fully connected layers, while the decoder combines a robot dynamic model and data-driven residual model. The decoder is solely used in training for reconstruction purposes, while a classifier leverages the latent variable for fault diagnosis both in training and inference. Fault cases of increased joint friction, partial loss of actuator effectiveness, and drivetrain mechani- cal faults are considered. In ablation studies, the approach performed superiorly to non-PIML vari- ants, evaluated with both a simulated UR5 dataset and a real industrial robot in-situ dataset. 4.5.3 Health Assessment An overview of all studies (18) employing learn- ing bias for health assessment is provided in Table 15. Focusing on SOH estimation in lithium- ion batteries, Y. Deng et al. (2025) train an N on features extracted from charge, discharge, and incremental-capacity curves. An additional loss term penalizes deviations from a monotonic relationship between the peak of the incremental- capacity curve and SOH, reflecting their consis- tently one-directional trend over aging. Trained on two public aging datasets with different chemistries and operating conditions, the result- ing model achieves lower SOH estimation errors than a standard N and a CNN. The challenge of accurately estimating the SOH of lithium-ion batteries under dynamic operating conditions is studied by F. Wang, Wu, et al. (2024). Building on ResNet, the authors use a constraint that guides relative distances and ranking between the embed- ding space and the output space (SOH) as prior knowledge. This battery degradation property was integrated by means of the Rank-N-Contrast loss. Validated on two datasets, the approach demon- strates superior performance in terms of predictive accuracy and structured latent representations compared to traditional and ML-based methods. Additional aspects of this work that fall into the class of hybrid approaches are reported in Section 4.6.3. Navidi et al. (2024) compare four approaches for estimating the capacity (SOH) and three internal degradation modes of lithium-ion bat- teries. As a follow-up study to Thelen et al. (2022), it includes three approaches that incor- porate observational bias, which are described in Section 4.3.3. Furthermore, the authors intro- duce an approach in which a shallow N predicts half-cell model parameters. A differentiable sur- rogate of the half-cell model is embedded via additional loss terms that weakly enforces con- sistency of the predicted parameters and the degradation behavior implied by the half-cell model. Trained on early-life experimental data together with simulation data, the regularized net- work outperforms an otherwise identical purely data-driven network and all approaches incor- porating observational bias. Explicitly account- ing for electrochemical parameter inconsistencies across cells, S. Zhang, Liu, Xu, Chen, and Su (2025) present an approach to estimating the SOH of lithium-ion batteries. Dedicated subnetworks jointly estimate lithium-ion concentration dynam- ics and cell-specific electrochemical parameters by processing initial-state features from an early-life discharge to encode parameter variability, and a capacity-difference sequence and sampled time coordinates to capture degradation behavior. The estimated internal states and parameters are then passed through a reduced electrochemical aging model (enhanced Single-Particle Model (SPM) with polynomial solid-phase diffusion, Butler- Volmer kinetics, Pad Ěe-approximated electrolyte diffusion, and electrode stoichiometry shifts) to reconstruct terminal voltage and capacity, with its governing equations and boundary conditions enforced via additional loss terms during training. Across multiple battery chemistries and operating conditions, the method attains very low SOH- prediction errors even with scarce training data and supports fast inference, outperforming base- lines such as PINN, CNN, and enhanced SPM, though at the cost of higher training complexity. S. Singh et al. (2023) address the joint estima- tion of SOC and SOH for lithium-ion cells operat- ing under varying temperatures and limited mea- surement data (see Fig. 7). Their method incor- porates the PDE governing solid-phase lithium diffusion from an SPM, along with Neumann 36 Table 15All studies employinglearning biasto addresshealth assessment, listed in alphabetical order. Learning Bias for Health Assessment ReferencePrior Physical KnowledgeUse Case TypeRepresentation Y. Deng, Du, and Ren (2025) Empirical aging trendMonotonicity con- straint (loss term) Lithium-ion bat- tery Freeman, Tang, Huang, and Vanzwieten (2022) Turbulence intensity as analyti- cal and empirical definition Algebraic equationOcean current turbines Jang et al. (2025)Lumped energy-balance model for battery heat generation First-order ODELithium-ion bat- tery S. Li et al. (2025)Empirical flank tool wear model Algebraic equationMilling Y. Liu et al. (2025)Mechanistic SEI-growth capacity-fade model Nonlinear ODELithium-ion bat- tery Navidi et al. (2024)Half-cell degradation modelAlgebraic equationsLithium-ion bat- tery Y. Pan, Sharif Khodaei, and Aliabadi (2025) Parisâ lawODEFatigue crack propagation (alu- minum specimens) S. Singh, Ebongue, Rezaei, and Birke (2023) Fickâs second law of diffusion, initial/boundary conditions PDE, algebraic equations Lithium-ion bat- tery H. Sun et al. (2022)Analytical relationships between signal-derived features and crack geometries Closed-form algebraic equations and inequal- ity constraints Microcrack quan- tification in aluminum plate Tefera, Van Baelen, Meire, Luca, and Kars- makers (2025) Monotonicity assumption, boundary constraint Algebraic equationBearing L. Wang, Yang, and Hu (2025) Semi-empirical Verhulst degra- dation model with Arrhenius temperature factor Nonlinear logistic-type ODE Lithium-ion bat- tery Wang, Z., Zhou, Xu, Sun, and Yan (2023) Discharge pressure model describing leakage First-order ODEAxial piston pump F. Wang, Wu, et al. (2024) Continuous degradation-trendRank-N-contrast lossLithium-ion bat- tery F. Wang, Zhai, Zhao, Di, and Chen (2024) Monotonic degradation, multi- variate degradation trend Regularization term, learnable dynamic model for PDE loss Lithium-ion bat- tery F. Wang, Zhai, Di, Zhao, and Chen (2025) Multivariate degradation trendLearnable dynamic model for PDE loss Lithium-ion bat- tery Wen et al. (2023)Semi-empirical Verhulst degra- dation model Nonlinear logistic-type ODE Lithium-ion bat- tery W. Xu, Zhou, et al. (2024) Pressure pulsation response model Algebraic equationGear pump S. Zhang, Liu, Xu, Chen, and Su (2025) Electrochemical aging modelCoupled ODEs, alge- braic equations Lithium-ion bat- tery flux boundary conditions driven by the applied current, into the loss function. To balance the data and physics-informed loss terms, the authors employ gradient normalization for adaptive loss balancing based on the work of Z. Chen, Badri- narayanan, Lee, and Rabinovich (2018), although no further details on its implementation are given. The PINN predicts spatio-temporal lithium con- centration fields within the anode, from which 37 Hybrid Modeling Of Lithium-Ion Battery: Physics-Informed Neural Network for Battery State Estimation Singh et al., 2023 Use Case:Lithium-ion battery health assessment Prior Knowledge:Fickâs law of diffusion â˘Li-ion transport inside the solid electrode particles is modeled by Fickâs second law of diffusion in spherical coordinates: , withli-ion concentration, solid-phase diffusion coeff.,radial coordinate andtime â˘Initial and Neumann boundary conditions Incorporation:PINN â˘The PDE for solid-phase diffusion is enforced through a physics residual loss â˘Automatic differentiation is used to obtain the derivatives â˘The residual for Fickâs law is calculated â˘The accompanying conditions are imposed as separate losses â˘Composite Loss:Loss = 1 Loss data + 2 Loss physics , with 1 , 2 loss weights N r t c Fickâs Law Initial Condition Data LossPhysical Loss Total Loss Balancing Boundary Condition Fig. 7A representative example of learning-bias approaches is proposed by S. Singh et al. (2023), demonstrating how prior physical knowledge can be introduced in the loss function. Own illustration based on the corresponding study. See the original work for full technical details. SOC and capacity-based SOH are inferred via established concentration-to-SOC and capacity relations. Results on three cells cycled at differ- ent temperatures show that the method accurately tracks both SOC and SOH over time. Additionally, an ablation study on one cell, where the model is trained on early-life check-up cycles and then applied to subsequent cycles, provides a limited demonstration of short-horizon SOH prognosis. Given that the heat generation rate varies signifi- cantly with the SOH, a PINN is adopted by Jang et al. (2025), incorporating a lumped energy- balance equation into its loss function to infer both temperature and the time-varying heat gen- eration rate in a manner consistent with battery thermodynamics. Subsequently, three methods are proposed that leverage the resulting PINN-derived profiles to estimate the SOH: an MLP, a 1D-CNN, and an LSTM-CNN-based approach. However, the evaluation is limited to comparisons among these three methods, leaving their comparative perfor- mance relative to established electrical-based or data-driven SOH estimation approaches unclear. Wen et al. (2023) also adopt a PINN-based approach by embedding a semi-empirical Verhulst degradation model (Xian, Long, Li, & Wang, 2013) into the loss function of an N for lithium- ion battery SOH estimation. An uncertainty- based weighting scheme adaptively balances the losses during training, allowing the PINN to exploit the Verhulst model without manual tuning of loss weights. Experiments demonstrate that this approach achieves lower SOH estimation errors compared to both a purely data-driven counter- part and a GPR-based method. Adopting a similar approach, L. Wang, Yang, and Hu (2025) leverage the same Verhulst model (Xian et al., 2013) (while additionally integrating the Arrhenius equation) as a soft constraint in the loss function, resulting in a PINN-based approach. SOH labels are derived from multi-stage constant-current charging via an incremental capacity-based method. Driving- style-related features, together with key operating variables, are used by the PINN to estimate the SOH of lithium-ion batteries in electric vehicles. Compared with both purely data-driven baselines and alternative PINN-based approaches, the pro- posed model achieves the lowest SOH estimation errors. Addressing SOH estimation for lithium-ion batteries via capacity loss, Y. Liu et al. (2025) aim to obtain accurate and physically plausi- ble degradation trajectories from charge-discharge data. They first compute entropy-based health indicators from voltage and current, and use these features together with cycle count as inputs to a feedforward N. A mechanistic SEI-growth model 38 for capacity fade is then embedded as an addi- tional physics-based loss term that penalizes mis- matches between the networkâs time derivative of capacity loss and the SEI model, thereby regu- larizing the network toward smooth, mechanisti- cally consistent SOH evolution and suppressing non-physical fluctuations. Across three datasets with different chemistries and cycling regimes, this physics-regularized model achieves substan- tially lower errors than purely data-driven MLP- and CNN-based models. Also covering SOH esti- mation for lithium-ion batteries, F. Wang, Zhai, et al. (2024) employ a PINN-based approach, in which one N estimates the SOH based on the current cycle and features generated from sen- sor readings in the charge phase, and a second N estimates the battery degradation dynamic behavior solely needed for loss construction. From a PIML perspective, besides a monotonicity loss, a PDE-based loss is also integrated. The latter accounts for the requirement that the degradation trajectory depends on charging rate, discharging rate, temperature, etc., rather than being merely a time-dependent univariate function. In compar- ison with an MLP and a CNN, the proposed method performs better in terms of predictive performance, data efficiency and generalizability, as demonstrated in experiments covering TL sce- narios and data-sparse scenarios. Except for the monotonicity loss, the main aspects of this work are also adopted by F. Wang et al. (2025), where they are specialized for 2-minute data segments obtained from laboratory experiments simulating satellite batteries. Tefera et al. (2025) address the problem of loss balancing in regularization approaches. Con- sidering a convolutional AE, the latent variable is fed into multiple fully connected layers for extraction of a HI suitable for bearing health assessment. Three constraints are directly inte- grated into the gradient descent algorithm instead of loss terms. Besides a monotonicity constraint, a boundary constraint leading to a normalized HI and a constraint that guides consistency between the signal energy and the HI are employed. The approach shows better predictive performance in the majority of experiments compared to other convolutional AE baselines. In this study, S. Li et al. (2025) leverage an empirical tool flank wear model relating wear to milling time as prior knowledge for health assess- ment in high-speed milling. This knowledge is incorporated via a loss function that penalizes deviations between the model predictions and the empirical model during the training of a GRU. The regularization enforces physically consistent degradation behavior during training. As a result, the model achieves improved physical consistency while maintaining low prediction errors compared to baselines. See Section 4.4.3 for a description of the inductive bias incorporated by S. Li et al. (2025). Wang, Z. et al. (2023) incorporate a dis- charge pressure model for axial piston pumps as a dynamic loss into a PINN. Hence, geometrical parameters related to leakage that represent the current health state of the piston-cylinder inter- faces can be fitted. Determining these parameters is challenging using traditional methods. As the leakage of the aforementioned intersections can be quantified for multiple piston-cylinder pairs inde- pendently, this can also be considered a diagnosis approach. Studying gear pumps, W. Xu, Zhou, et al. (2024) tackle the black-box nature of DL models. The output of a compound N (consist- ing of CNN, BiLSTM and attention) is guided to match the parameters of a physical pressure model regarding the outlet pressure pulsation. The physics-based model can be used to recon- struct the input signal to perform unsupervised training. The estimated physical parameters are used to construct a health indicator based on distance metrics. The proposed method is supe- rior to purely data-driven approaches in terms of interpretability (the authors show this qualita- tively by means of t-distributed stochastic neigh- bor embedding), though not necessarily in terms of reconstruction capabilities. Y. Pan et al. (2025) address both the small- sample problem and the poor generalization abil- ity in ML-based fatigue crack quantification. Using aluminum specimens representative of air- craft structures, the authors model fatigue crack propagation with an LSTM network. A Paris law- based crack-growth model is utilized to formulate an additional loss term aiming to improve the pre- dictive accuracy for crack growth under complex environmental conditions. The positive contribu- tion of the incorporated physics to the modelâs overall performance is demonstrated through ded- icated ablation studies. With the incorporated 39 observational bias described in Section 4.3.3, H. Sun et al. (2022) also introduce a learning bias by augmenting the training objective with additional physics-based loss terms that penalize inconsistencies between the network outputs and analytical relationships derived from guided-wave scattering theory. For crack length, an additional loss term penalizes pairs of samples whose pre- dicted lengths violate the required monotonic relationship between a width-like feature (con- structed from neighboring reflection amplitudes) and the angular spread of the reflected lobe. For depth and direction, further penalty terms enforce analytical formulas that link the ratio of reflected to transmitted energy and the corrected reflec- tion angle, respectively, to the corresponding crack parameters given the current length prediction. A single weighting factor controls the influence of all physics-based penalties; sensitivity studies show that choosing this weight appropriately yields sub- stantially lower quantification errors than training without these constraints. The method of Freeman et al. (2022) enables the classification of rotor blade pitch imbal- ance faults in ocean current turbines according to their level of severity. An ML pipeline with the single-phase power output of the turbine as input consists of principal component analysis and multinomial logistic regression for flow-speed classification, an N with an augmented loss for turbulence-intensity classification and a final N for fault severity classification. Both an empirical and an analytical expression of turbulence inten- sity are used to define a monotonicity constraint as an additional loss. In ablation studies, the pro- posed method outperforms its purely data-driven counterpart in terms of predictive performance. 4.5.4 Prognosis Thirteen studies employ learning bias for prog- nosis (see Tab. 16), several of which focus on lithium-ion battery prognostics. SOH prognosis of lithium-ion batteries is addressed by Z. Xu et al. (2022). A ResNet-based encoder-decoder model uses capacity and secondary variables (i.e., temperature, voltage, and current) from the cur- rent cycle to predict capacity and full secondary- variable profiles for the subsequent cycle. This one-step-ahead predictor is iterated from the first discharge cycle to generate long-horizon degrada- tion trajectories over the batteryâs life. Physics is incorporated through additional loss terms that penalize violations of a first-order Thevenin-based state equation and a capacity-balance equation, which are combined into an ODE system. The pro- posed method achieves substantially lower SOH prediction errors than several GP-based baselines using only the first discharge cycle, with abla- tion studies attributing the performance gains to the added regularization based on the underlying battery model. Pugalenthi et al. (2024) present a prognostic framework for lithium-ion batter- ies, where an N is first identified from a single run-to-failure cell using a PF and then used to predict the capacity trajectories of other cells. A semi-empirical SEI-film-based capacity fade model is incorporated as an additional loss term to improve physical plausibility and reduce errors in predicted capacity, especially when only lim- ited training data are available. The trade-off is higher computational cost due to the extra over- head of evaluating the physics-based loss on top of the PF-based parameter estimation. Z. He et al. (2025) propose a degradation modeling frame- work that combines DL with a Wiener process for lithium-ion batteries. The network takes degra- dation features (i.e., constant-current charging time, incremental-capacity peak, and tempera- ture peak) and maps them through intermedi- ate physical variables that represent impedance- related quantities, before producing a nonlinear degradation path that serves as the drift of the Wiener process. A composite loss combining mean squared error on these intermediate physical vari- ables with a likelihood-based term for degradation increments is used to train both the network and stochastic-process parameters jointly from historical data. The latter are further updated online via Bayesian inference using real-time SOH measurements. Ablation and comparative stud- ies on two battery datasets show that adding the impedance-informed latent layer and the Wiener process module improves not only SOH trajectory prediction but also the fidelity of the resulting reliability curves and RUL and lifetime distri- butions, compared with purely data-driven and purely stochastic-process baselines. To effectively 40 Table 16All studies employinglearning biasto addressprognosis, listed in alphabetical order. Learning Bias for Prognosis ReferencePrior Physical KnowledgeUse Case TypeRepresentation Badora et al. (2023)Fracture-mechanics relation between load ratio of ther- mal stresses and corresponding stress intensity factors Algebraic ratio con- straint High-pressure nozzle of an industrial gas tur- bine E et al. (2025)Empirical aging law relating SOH to cycle number Algebraic equationSupercapacitors Fassi, Heiries, Boutet, and Boisseau (2024a) Empirical aging trendMonotonicity and boundedness con- straints (loss terms) Metal-oxide- semiconductor field-effect transis- tor Z. He et al. (2025)Impedance-based degradation mechanism, stochastic degrada- tion behavior (Wiener process) Intermediate physi- cal variables, Gaussian increment model Lithium-ion bat- tery Najera-Flores, Hu, Chadha, and Todd (2023) Empirical aging trendMonotonicity con- straint (loss term) Lithium-ion bat- tery Pugalenthi, Park, Hus- sain, and Raghavan (2024) Semi-empirical SEI-film-based capacity fade model Two-term exponential equation Lithium-ion bat- tery Ramirez et al. (2024)Heat-diffusion model of transformer oil, standard thermal-aging model for wind- ing insulation 1D heat-diffusion PDE, algebraic equations Transformer Y. Wang et al. (2024)Monotonicity assumptionSingle-term exponential function Bearing, trans- former, electrome- chanical servo system E. Wang, Zhou, Wen, Liu, and Chen (2025) Koopman operator theoryLinear operator with algebraic consistency loss Turbofan engine, bearing L. Wang, Wang, Yang, and Liao (2025) Reliability model of bearing failure process Weibull cumulative dis- tribution function Bearing Z. Xu, Guo, and Saleh (2022) First-order Thevenin modelODEsLithium-ion bat- tery S. Zhang, Liu, Xu, Guo, and Su (2025) Enhanced SPMODE system with alge- braic voltage relation Lithium-ion bat- tery Y. Zhu, Cheng, et al. (2024) Inverse monotonic relationship between crack surface area and RUL Monotonicity con- straint (loss term) Bearing target RUL prediction of lithium-ion batteries, S. Zhang, Liu, Xu, Guo, and Su (2025) pro- pose a two-stage approach. In the first stage, an electrochemical-informed generative model, con- strained by a reduced-order enhanced SPM, recon- structs electrode-level states. The model is trained with a composite loss over the electrochemical governing equations, initial and boundary con- ditions, and terminal-voltage mismatch, whose weights are adaptively balanced via gradient-norm ratios. In the second stage, incremental-capac- ity and differential-voltage curves derived from 41 the reconstructed states serve as electrode-lev- el features. These are combined with cell-level features (capacity-difference curves and charg- ing protocols) in an N: two CNN branches encode the cell-level inputs, while a GRU with self-attention encodes the electrode-level inputs, and the concatenated representations are decoded to predict the RUL. Across four datasets with different chemistries and operating conditions, the method outperforms both mechanistic and purely data-driven baselines in terms of prediction error, shows better robustness with limited train- ing data, and offers faster inference, while also enabling identification of electrode-level degrada- tion modes. Early-stage RUL prediction is studied by Najera-Flores et al. (2023). To this end, a neu- ral differential operator is learned for the discharge capacity rate from early-life cycling data. The pro- posed architecture draws inspiration from Deep- ONet (L. Lu, Jin, Pang, Zhang, & Karniadakis, 2021), featuring a Bayesian branch network that encodes cell-specific early-life features and a deter- ministic trunk network that encodes time. Multi- ple loss terms are employed to promote accurate modeling of the discharge capacity, including a monotonicity constraint that weakly enforces neg- ative self-acceleration of the capacity trajectory. At inference, the learned operator is sampled from the Bayesian branch, integrated forward in time to reconstruct the capacity curve, and the RUL is obtained as the difference between the predicted end-of-life time (at a capacity threshold) and the current cycle. Experiments show that, given the same early-life training data, the proposed physics-constrained Bayesian operator achieves smaller RUL prediction errors and more reli- able uncertainty quantification than a simplified physics-based failure forecast model, a GPR-based method, and a similarity-based method. L. Wang, Wang, et al. (2025) propose an approach for bearing RUL prediction using acous- tic emission signals. A novel health indicator is introduced to quantify acoustic emission signal complexity and is shown to outperform standard time-domain and entropy features in monotonic- ity, robustness, and trendability. This health indi- cator is fed to an LSTM whose loss combines mean squared error with a Weibull-based term derived from reliability engineering, thereby con- straining the learned degradation trajectory to be consistent with the expected failure behav- ior. Experiments demonstrate that the proposed method yields substantially lower RUL prediction errors than a conventional LSTM. Already dis- cussed in Section 4.3.4, Y. Zhu, Cheng, et al. (2024) propose three mechanisms for incorporat- ing physics to tackle bearing RUL prediction. In terms of learning bias, the authors design an inconsistency loss that penalizes pairs of pre- dicted RUL values violating the inverse monotonic relation between crack surface area and RULâ imposed via a ReLU-based penalty. Again, the complete framework proves superior, but without ablation experiments, the specific contribution of the additional loss term cannot be quantified. See Section 4.6.4 for a synopsis focusing on the hybrid aspect of the framework. To facilitate prognostics of complex industrial machinery, E. Wang et al. (2025) present a novel approach grounded in Koopman operator the- ory. The proposed Koopman-informed N enables accurate RUL prediction by learning nonlinear system dynamics through a linear representa- tion in a latent eigenfunction space. Within an encoder-decoder architecture, the encoder maps data spanning the entire operational lifespan into an eigenfunction space, where the systemâs evo- lution is approximated by a finite-dimensional Koopman matrix: multi-step temporal forecast- ing is performed via repeated applications of the learned forward and backward Koopman oper- ators. The decoder reconstructs future system states from the propagated latent representa- tion. Multiple loss terms (including a consistency loss that penalizes discrepancies between forward and backward evolution) regularize the learned dynamics and promote stable, physically coherent temporal behavior. Lastly, a nonlinear regression head leverages the learned high-level features to produce precise RUL estimates. The proposed Koopman-informed N is evaluated on both a bearing and a turbofan engine dataset, where the model consistently outperforms several strong DL baselines in terms of predictive performance. Y. Wang et al. (2024) propose a Transformer- based prognostics model designed for scenar- ios lacking reliable degradation physics. The approach hinges on decomposing time-series data into a slowly varying trend component (captur- ing the overall degradation trend) and a residual 42 component (capturing higher-frequency fluctua- tions). Thereafter, a simple monotonic paramet- ric curve is fitted to the trend component in a sliding-window procedure. Rather than construct- ing accurate, domain-specific degradation models, these local fits (here, single-term exponentials) are treated as âsimple, general and imperfectâ approximations. An additional loss term penal- izes large deviations from these fitted curves, thereby regularizing the model toward predictions that reflect the assumed irreversibility of degrada- tion. Given its domain-agnostic prior knowledge, the model is applied to three different use cases (bearings, transformers, and an electromechani- cal servo system), where it consistently reduces long-horizon prediction errors compared to several state-of-the-art DL models. Given that failure of power semiconductor devices poses a significant reliability challenge for power converter systems, Fassi et al. (2024a) address RUL prediction for power metal-oxide- semiconductor field-effect transistors under ther- mal aging. To improve predictive accuracy and physical consistency, the authors incorporate addi- tional loss terms that weakly enforce monotonic, bounded RUL trajectories via ReLU-based penal- ties. A series of experiments across various recur- rent network architectures demonstrates that it achieves lower or comparable mean squared error than purely data-driven counterparts, while simul- taneously enabling faster convergence. E et al. (2025) present a method for predicting the RUL of commercial supercapacitors by embed- ding an empirical aging law that models SOH as a logarithmic function of cycle number into the loss function of an LSTM. A scalar weighting factor between the data and physics losses is tuned via Bayesian optimization, leading to stronger regu- larization under scarce data and reduced reliance on the physical loss term as more data become available. Experiments demonstrate that the pro- posed model substantially outperforms its purely data-driven counterpart, with prediction errors comparable to more advanced data-driven meth- ods while using only a fraction of the full life cycle as training data. In addition to the inductive bias reported in Section 4.4.4, Badora et al. (2023) incorpo- rate a learning bias independent of the former. An additional loss term regularizes training with respect to the fracture-mechanics load-ratio rela- tion between applied thermal stresses and the cor- responding stress intensity factors, using a large set of synthetically generated stress-crack-length combinations. A dynamically adjusted weighting factor gradually shifts emphasis from this physics term toward the empirical crack-length error as training progresses, ensuring that the final model both respects the underlying fracture mechanics and fits the sparse inspection data. While the need for a dynamic weighting between physics and empirical loss terms is well motivated, the specific piecewise schedule for the weighting coefficient is introduced without justification, making this part of the approach heuristic and potentially hard to generalize or reproduce. Combined with the induc- tive encoding of Parisâ law, the proposed method effectively targets fatigue crack growth modeling in a gas turbine nozzle. Ramirez et al. (2024) propose an efficient spatio-temporal model for predicting transformer winding temperature, including the local hotspot, and insulation aging. Embedding a simplified one-dimensional heat-diffusion PDE with uniform heating into a PINN enables estimating oil tem- peratures, which are then used to calculate wind- ing temperatures, aging acceleration factors, and the associated loss of life. Tested on a distribution transformer in a floating photovoltaic power plant, the method closely matches numerical PDE solu- tions and improves hotspot and aging estimation compared to a standard analytic hotspot model, with validation against fiber optic sensor mea- surements. Additionally, a residual-based atten- tion scheme improves convergence and training stability of the PINN. 4.6 Hybrid Approaches Hybrid approaches are characterized by combining independent physics-based and data-driven mod- els, either in parallel or in series. The identified studies are organized by PHM task in Tables 17â 20. A representative example is illustrated in Figure 8. 4.6.1 Fault Detection Two studies employing hybrid approaches for fault detection have been identified (see Tab. 17), both targeting lithium-ion batteries. Firoozi et al. 43 Cylindrical Battery Fault Detection Under Extreme Fast Charging: A Physics-Based Learning Approach Firoozi et al., 2022 Use Case:Lithium-ion battery fault detection Prior Knowledge:Electrochemical and thermal dynamics â˘Electrochemical model: anode SPM lithium-diffusion model for terminal voltage (); state space representation: ⢠Ⲡ, âł â˘Thermal model: radial energy-balance heat-diffusion model for cell temperature (); state space representation: ⢠ⲠⳠ, Incorporation:In-parallel coupling (hybrid) â˘Two GPR models use current, voltage and temperature to learn the nonlinear uncertainty corrections and â˘Adding and to the physics-based model outputs and makes the mismatches and largely reflect the fault terms âł and âł , rather than modeling/measurement errors Experimental Data Thermal Model Fault / No Fault State Space Decision Maker (Threshold-based) Electrochemical Model State Space GPRGPR Fig. 8A representative example for hybrid approaches is proposed by Firoozi, Sattarzadeh, and Dey (2022), demonstrating how physics-based and ML models can be combined. Own illustration based on the corresponding study. For brevity, time indices are omitted, with the prime denoting the next state. See the original work for full technical details. Table 17All studies employinghybrid approachesto addressfault detection, listed in alphabetical order. Hybrid Approaches for Fault Detection ReferencePrior Physical KnowledgeUse Case TypeRepresentation Firoozi et al. (2022)SPM model and radial thermal model Reduced-order ODE state-space model Lithium-ion bat- tery L. Zhang, Xia, and Zhang (2024) Second-order ECM coupled with a lumped thermal model ODE state-space model, lumped ODE Lithium-ion bat- tery (2022) target real-time fault detection in cylin- drical lithium-ion batteries under extreme fast charging, aiming for early detection of voltage and thermal faults (see Fig. 8). Two detection observers are built on an experimentally identi- fied reduced-order electrochemical-thermal model; a GPR is used in parallel to learn additive voltage and temperature uncertainty terms from residu- als between model predictions and measurements in no-fault cycles. The learned uncertainty correc- tions are fed back into the observers to suppress non-fault deviations, and residuals are evaluated against calibrated thresholds to detect voltage and thermal faults. Comparative results with a model- only observer indicate that incorporating the GPR can enable detection of smaller faults that the purely physics-based counterpart misses. Building on a similar concept, L. Zhang et al. (2024) (who also reference Firoozi et al. (2022) as related work) propose an adaptive fault detection framework for lithium-ion batteries that combines a thermoelec- tric model-based EKF observer with a BiLSTM. A second-order ECM coupled with a simplified thermal model is used in the EKF to estimate volt- age, temperature, and SOC, while the BiLSTM is trained on healthy data to learn the voltage obser- vation error and subsequently compensates the observer to suppress uncertainty-induced residu- als. The corrected residuals are compared against a threshold calibrated from healthy operation to detect soft internal short circuit faults. Experi- ments on driving-cycle data show that the pro- posed framework yields substantially lower volt- age estimation errors than the standalone EKF, thereby enabling more reliable fault detection with fewer false alarms. 44 Table 18All studies employinghybrid approachesto addressdiagnosis, listed in alphabetical order. Hybrid Approaches for Diagnosis ReferencePrior Physical KnowledgeUse Case TypeRepresentation Pettorossi, Morvillier, Heiries, Rosini, and Ger- ard (2025) Proton exchange membrane fuel cell simulation ODE systemFuel cell S.K. Singh et al. (2024)0D high-fidelity physics-based engine model Coupled algebraic equations, ODEs Diesel engine J. Xu et al. (2023)Fault hierarchies and statisti- cal fault evolution/propagation mechanisms (time-to-failure behavior and correlations between failure modes) Probability density functions and cumu- lative distributions of failure modes, algebraic update formulas Offshore wind tur- bine 4.6.2 Diagnosis Table 18 lists the three studies that employ hybrid approaches for diagnosis. Pettorossi et al. (2025) address fault diagnosis for proton exchange mem- brane fuel cells, focusing on identifying and iso- lating four different fault types: flooding, drying, air starvation, and hydrogen starvation. To tackle this, the authors propose an approach that com- bines a physics-based proton exchange membrane fuel cell model with an LSTM. The physics-based model provides estimates of unmeasured process variables, including membrane resistance, water content, and current density distribution. These variables are combined with measured stack sig- nals and fed to the LSTM, both in training and inference. Hence, more informed and robust fault classification is enabled. This integration improves diagnostic accuracy, reduces detection time, and enhances generalizability compared to purely data-driven approaches. S.K. Singh et al. (2024) propose a fault diag- nosis framework for a diesel engine that com- bines a 0D (lumped, time-dependent) high-fidelity physics-based engine model with an AE and a 1D-CNN. The AE compresses high-dimensional sensor data into a latent feature vector, while the physics-based engine model provides additional simulated variables. These are then concatenated and passed to a 1D-CNN that classifies four con- ditions: nominal operation, and faults due to injection pressure, injection duration, and start of injection. The engine model thus acts as an in- parallel source of physics-based features that com- plement the data-driven latent representation for fault classification. Experiments on test-bed data show that the proposed model achieves higher diagnostic accuracy than a purely data-driven 1D- CNN and exhibits improved robustness to sensor noise and to extrapolation across unseen engine speeds and operating conditions, with notably fewer false positives in nominal conditions. Fault root cause tracing in complex electrome- chanical systems is studied by J. Xu et al. (2023), demonstrated on an offshore wind turbine expe- riencing an unscheduled power drop. Common physics- and statistics-based knowledge about fault mechanisms is first encoded into a static hierarchical fault root cause tracing networkâa probabilistic graph whose nodes and edges rep- resent functional units, fault modes, and their propagation relationships. This model is then refined with operation data: anomalies detected via a Wasserstein GAN, together with statistical laws describing how faults evolve over operat- ing time, update the weights of fault nodes and edges. Finally, a bidirectional probabilistic reason- ing scheme combines forward fault propagation and backward tracing information across the hier- archy to rank fault nodes and identify the most likely root cause and fault paths. 4.6.3 Health Assessment Table 19 summarizes the five studies that employ hybrid approaches for health assessment, all but one of which target lithium-ion batteries. Address- ing SOH estimation, X. Feng, Zhang, Xiong, and Wang (2024) leverage a second-order resistor- capacitor ECM, whose parameters are identified from post-charge relaxation voltage via nonlin- ear least squares and then used as inputs to 45 various ML models, including GPR, XGBoost, SVR, elastic net, and a simple MLP. For each regressor, the hybrid approach is compared with its purely data-driven counterpart trained either on raw relaxation voltage samples or on statisti- cal features extracted from the relaxation curve. The results indicate that the hybrid GPR gen- erally achieves the lowest prediction error, with its advantage particularly evident when training data or relaxation time are limited. Essentially following the same approach, Lin et al. (2025) use a fractional-order ECM, whose parameters are identified via recursive least squares and then passed to an RF regressor for SOH estimation. Although not investigating further ML models, they also benchmark the hybrid RF against a stan- dard RF (either trained on raw relaxation data or on statistical features), with the former generally outperforming its counterparts in terms of pre- diction error. Kohtz and Wang (2022) present a hybrid approach for the online joint SOC and SOH estimation of lithium-ion batteries. A dual EKF is built on a simple empirical voltage measure- ment function (polynomial open-circuit voltage- type relation in SOC, capacity, and current), and an N is trained offline as a residual model to learn the error between this physics-based mea- surement function and the measured voltage. In online operation, the N correction is added to the empirical measurement function within the dual EKF measurement equation. Results show that embedding the residual model significantly reduces capacity estimation error, demonstrating more accurate battery health assessment. In addi- tion to the learning bias discussed in Section 4.5.3, the method proposed by F. Wang, Wu, et al. (2024) also falls into the class of in-series hybrid approaches. An ECMâs identified parameters are concatenated with data-driven features extracted from current measurements, both in training and inference. The concatenated feature vector forms the input for a subsequent encoder. W. Xu, Wang, et al. (2024) present a digi- tal twin-based framework for wear state assess- ment of gear pumps in fuel control systems. A first-principles dynamic model of the fuel sys- tem is built in Simulink, while a deep RL agent adaptively updates flow correction coefficients so that simulated pressures match measured ones. These learned coefficients, together with normal- ized operating conditions and model error, form an interpretable wear feature vector whose distance to a healthy reference indicates wear severity. 4.6.4 Prognosis A total of ten studies employ hybrid approaches for prognosis, as listed in Table 20. B. Sun et al. (2023) propose a prognostics framework for lithium-ion batteries, in which a multi-physics simulation model (electrochemical-thermal with SEI-coupling) and an LSTM operate in series. The simulation model uses measured current, voltage and temperature to estimate SOH, which is then used both as training targets and as a slower, high-fidelity reference to periodically recalibrate the LSTM during operation. The LSTM employs a dynamically sized input window whose length is adapted based on the Kullback-Leibler divergence between consecutive windows, so that the net- work alternately emphasizes long-term trends and short-term fluctuations. An adaptive evolution mechanism retrains and updates the LSTM when- ever its predictions deviate too much from the simulation, improving long-term prediction per- formance. Compared to a conventional LSTM, the proposed framework consistently achieves lower RUL prediction errors across two datasets and laboratory experiments conducted under vary- ing operating conditions. Specifically tailored for data-scarce scenarios, Liang et al. (2024) propose a hybrid method for RUL prediction and uncer- tainty quantification in lithium-ion batteries. An ensemble learning approach is adopted to inte- grate an empirical degradation model and a data- driven component. While the former, a double exponential degradation model, captures the non- linear degradation trend, the GRU-CNN network learns to predict short-term fluctuations. The out- put of these models, along with the preprocessed data, is fed into a Bayesian N that predicts RUL and provides uncertainty quantification, show- ing comparable performance to several alternative approaches and increased data efficiency in abla- tion studies. L. Ma et al. (2024) present an approach for predicting battery RUL from a sin- gle constant-current charging curve. The method also addresses the black-box nature of data-driven methods by combining battery physics and ML. Characteristic peaks from incremental capacity curves are approximated with Lorentzian func- tions and integrated to form a smooth, parametric 46 Table 19All studies employinghybrid approachesto addresshealth assessment, listed in alphabetical order. Hybrid Approaches for Health Assessment ReferencePrior Physical KnowledgeUse Case TypeRepresentation X. Feng et al. (2024)Second-order RC ECMNonlinear algebraic expressions of expo- nentials (derived from first-order linear ODEs) Lithium-ion bat- tery Kohtz and Wang (2022) Empirical voltage-SOC-capacity relation (open-circuit voltage- type measurement model) Algebraic equationLithium-ion bat- tery Lin et al. (2025)Fractional-order ECMFractional-order state- space model Lithium-ion bat- tery F. Wang, Wu, et al. (2024) Second-order RC ECMODE systemLithium-ion bat- tery W. Xu, Wang, et al. (2024) System-level lumped-parameter dynamic model Coupled nonlinear ODEs Gear pump capacity-voltage model. From this model, the peak centers, widths, and areas are extracted as fea- tures sensitive to degradation for a lightweight N that maps them to the RUL. Across dif- ferent chemistries and operating conditions, the proposed approach provides more stable and accu- rate predictions than purely data-driven methods. The framework of L. Xu et al. (2023) enables predicting lithium-ion battery degradation trajec- tories and RUL. Early-life features are extracted in three forms: the variance of the difference of discharge capacity-voltage curves between cycles 10 and 100; the anode state-of-lithiation change between cycles 90 and 100 obtained from a pseudo- 2-dimensional electrochemical model; and a final feature that multiplies the aforementioned to cap- ture both observable and internal degradation signals. Battery cells are clustered using k-means clustering based on the third feature mentioned to group similar aging patterns. Data augmentation techniques are applied to enrich cluster-specific datasets. For each cluster, an LSTM encoder- decoder model predicts the full capacity degra- dation trajectory from limited early cycles. The proposed method enables accurate and early pre- diction of battery life across different aging con- ditions and chemistries and performs better than GPR, SVR and an autoregressive LSTM baseline. Shi et al. (2022) present a physics-informed frame- work for lithium-ion battery degradation estima- tion and RUL prediction. Their method com- bines a calendar-and-cycle-aging model, expressed through five semi-empirical operating stress-factor formulations, with an LSTM. The physics-based component represents degradation associated with operating conditions (including cycle duration, rest time, temperature, SOC, and load), while the LSTM uses this modeled degradation together with cycle-level monitoring signals to learn degra- dation behavior that is not captured by the former alone. The estimated degradation trajectory is then passed to a separate LSTM that forecasts future capacity loss; RUL is obtained from the predicted point at which the end-of-life thresh- old is reached. In the reported experiments, the method achieves better capacity-fade modeling performance than the CNN and BiLSTM baselines under the tested conditions. Introducing three distinct mechanisms to inform a BiLSTM, Y. Zhu, Cheng, et al. (2024) propose a framework for bearing RUL predic- tion. One of these mechanisms represents an in- series approach: a parametric physics degradation model generates an explicit degradation trajec- tory, which is subsequently passed to the recurrent model to guide training by a mechanistic represen- tation of health progression. Combined with the observational and learning biases (see Sec. 4.3.4 and Sec. 4.5.4, respectively), the framework con- sistently outperforms purely data-driven baselines in terms of RUL prediction errors. Yet the lack of ablation studies prevents determining which of the three mechanisms yields the greatest benefit. 47 Table 20All studies employinghybrid approachesto addressprognosis, listed in alphabetical order. Hybrid Approaches for Prognosis ReferencePrior Physical KnowledgeUse Case TypeRepresentation Aizpurua, Knutsen, Heimdal, and Vanem (2023) Arrhenius-based thermal-stress model with Minerâs rule for sta- tor winding insulation Algebraic equationElectric motor Kundu, Darpe, and Kulkarni (2024) Pit-growth model inspired by Parisâ law Algebraic equationGearbox T. Li (2024)Fatigue crack growth model (Parisâ law) Algebraic equationFatigue crack growth in alu- minum lug joint Liang, Liu, and Xiao (2024) Double exponential model for capacity prediction Algebraic equationLithium-ion bat- tery L. Ma et al. (2024)Incremental capacity curves expressed as sum of Lorentzian functions Algebraic equationLithium-ion bat- tery Shi, Rivera, and Wu (2022) Semi-empirical calendar and cyclic aging model Algebraic equationLithium-ion bat- tery B. Sun et al. (2023)Electrochemical-thermal-SEI model Coupled PDEs plus SEI capacity-fade ODE, solved as a high- fidelity numerical simulation Lithium-ion bat- tery L. Xu, Deng, Xie, Lin, and Hu (2023) Pseudo-two-dimensional electrochemical model (Doyle- Fuller-Newman model) PDELithium-ion bat- tery K. Zhu, Guo, Li, and Lin (2024) Flank wear modelAlgebraic equationHigh-speed milling Y. Zhu, Cheng, et al. (2024) Vibration degradation modelAnalytic model defined by algebraic equations Bearing K. Zhu et al. (2024) target tool wear moni- toring and RUL prediction in high-speed milling. A BiLSTM-based architecture processes cutting force signals to extract temporal features, while a physics-based wear model (driven by the milling parameters) runs in parallel and provides a flank- wear estimate that conditions an attention mech- anism aggregating the learned features into a shared representation used for prediction. Assum- ing that tool wear and RUL are two manifestations of the same underlying tool state, the network jointly predicts both quantities from this shared representation under a single multi-task loss with two separate regression heads. Experiments over different milling conditions show that the pro- posed model yields clearly improved predictive accuracy compared with an otherwise identical purely data-driven model. To address fatigue crack prognosis in an alu- minum lug joint monitored by Lamb waves, T. Li (2024) targets RUL prediction within a hybrid, PF-based framework. Based on Parisâ law, a non- linear state-space model is constructed to repre- sent the evolution of fatigue crack length. Addi- tionally, a GPR model maps the Lamb-wave fea- ture to lifetime percentage, which is subsequently used to modify PF state and parameter sam- ples (prognostic-aided model updating) so that they are consistent with both past measurements and prognostic information. Applied to five spec- imens across five testing scenarios, the method yields more accurate RUL and lifetime-percentage predictions with generally tighter uncertainty bounds, and achieves more reliable prognosis than both the approach without prognostic-aided updating and the standalone GPR model. 48 In order to perform RUL prediction for gear- boxes, Kundu et al. (2024) propose multiple approaches. In one approach, Random Forest Regression (RFR) is used to determine the current pitting area based on a correlation coefficient- based HI constructed from both healthy and faulty gearbox vibration data. With the current pitting area known, a physical pit-growth model can be employed to estimate the number of remaining cycles until failure that directly relates to the RUL. Additionally, Bayesian inference techniques are used to update the parameters of the physical model during inference. Aizpurua et al. (2023) address prognostics of permanent magnet motors in a maritime con- text. With the winding insulation as the degra- dation quantity of interest, features derived from wind speed and vessel speed are used as inputs to two ML models connected in series. The first model predicts torque from operational and meteorological data, and the second predicts winding temperature using the predicted torque along with the same input features. The result- ing winding-temperature trajectory is then fed into an Arrhenius-based thermal-stress degrada- tion model with Minerâs rule and Monte Carlo simulation to obtain a probabilistic RUL esti- mate for the insulation. The authors benchmark several alternative ML models (including linear regression, gradient boosting, RF, and MLP) for the torque and temperature prediction tasks and select the best-performing configurations, but the hybrid RUL framework itself is only validated on a single case study and is not quantitatively compared to other prognostics approaches. 5 Discussion This section discusses the main findings of the review, focusing on recurring methodological pat- terns, the observed effects of incorporating physics into ML-based PHM, and implications for dif- ferent PHM tasks. It then highlights contextual limitations, barriers to deployment, and termino- logical issues in the current literature and con- cludes by outlining promising directions for future research. 5.1 Methodological Patterns Beyond the classification of individual methods (see Fig. 4), examining recurring methodological patterns provides insight into conceptual matu- rity of the field, the emergence of shared modeling principles, and the extent to which physics inte- gration has become methodologically standard- ized. The following analysis examines each class with respect to the diversity of implementation strategies, the strength of physics enforcement, and the implications for transferability and prac- tical adoption. 5.1.1 Observational Bias Methods incorporating observational bias rely on physics solely as a data source, while the model and training objectives remain conventional and agnostic to the underlying physics. Two dominant strategies emerge across the reviewed literature: using physics-based simulators to generate sim- ulated data that enrich limited or imbalanced datasets (Y. Deng et al., 2023; Kohtz et al., 2022; Y. Li et al., 2024; Mei et al., 2024; Y. Qin et al., 2024, 2025; Ren et al., 2025; Y. Zhu, Cheng, et al., 2024), and pretraining on simulated data followed by fine-tuning on scarce real-world data, typically in a TL setting (Y. Dong et al., 2022; S. Li et al., 2024; X. Liu et al., 2024; Y. Liu et al., 2024; Pet- torossi et al., 2024; Song et al., 2022; Y. Zhang, Feng, et al., 2023; Y. Zhang et al., 2024). Delta learning, a less common variant in which a model trained on simulated data is corrected by a sec- ondary model trained on experimental data, has also been explored (Navidi, Thelen, Li, & Hu, 2023; Thelen et al., 2022). This apparent methodological uniformity is not a sign of community consensus but rather a consequence of the definitional boundary. Because physics can enter only through additional training data, the set of feasible implementation strate- gies is inherently small, regardless of the specific PHM task or asset. This constraint simultaneously explains both the accessibility and the limita- tions of observational-bias approaches: they are straightforward to apply, provided that a simula- tor of sufficient fidelity can be constructed. Once training is complete, however, the simulator is dis- carded and the deployed model behaves like a conventional black box, retaining no mechanism to enforce physical plausibility at inference time. 49 Ultimately, observational-bias approaches are well-positioned to address data scarcity within regimes covered by the simulated data, yet struc- turally ill-suited to deliver on stronger promises such as physical consistency and reliable extrapo- lation to unseen regimesâproperties that require explicit structural or algorithmic enforcement. 5.1.2 Inductive Bias Methods incorporating inductive bias embed physical principles directly into the model through tailored architectural interventions. Consequently, they achieve the closest alignment between model structure and prior physical knowledge, albeit at the expense of being highly asset-specific. The resulting methodological landscape is corre- spondingly heterogeneous. Characteristic imple- mentation patterns include: custom recurrent cells informed by specific degradation laws (Badora et al., 2023; Dourado & Viana, 2019, 2022; Nasci- mento et al., 2021; Yucesan & Viana, 2019), graph topologies reflecting asset structure (Cheng et al., 2024; Y. Feng et al., 2022; Jin et al., 2024; S. Liu et al., 2025; M. Zhou et al., 2025), layers tailored to known frequency content (C. Gao et al., 2024; Zeng et al., 2025), informed GP priors (H. Huang et al., 2022; Qiang et al., 2023; K. Zhu et al., 2023), or activation and output constraints enforc- ing monotonicity (Hao et al., 2023; S. Li et al., 2025; C. Yin et al., 2025; Z. Zhou et al., 2023). While these patterns share broad conceptual goals, their implementations are fundamentally different and tightly coupled to the specific asset under study. Graph-based encodings naturally align with diagnostics in multi-component sys- tems (Y. Feng et al., 2022; Jin et al., 2024; S. Liu et al., 2025), whereas custom recurrent cells and informed GP priors predominantly serve prognos- tics (Badora et al., 2023; Yucesan & Viana, 2019; K. Zhu et al., 2023), where degradation dynam- ics must be captured temporally. Monotonicity constraints represent a notable exception: because they encode a domain-agnostic property of irre- versible degradation, they transfer readily across assets. Yet they provide only a weak inductive bias compared to the mechanistic models embedded in asset-specific architectures. Overall, research on inductive bias largely constitutes a proliferation of problem-specific solutionsâthe product of tailored model engi- neering rather than mature, reusable modeling strategies. While such tailoring yields strong per- formance and structurally enforces adherence to the embedded physical principles, it contributes little to developing reusable design patterns, leav- ing this class fragmented. 5.1.3 Learning Bias Learning-bias approaches are structurally more uniform, with physics incorporated almost exclu- sively as additional terms in the loss function, regularizing the model toward physically plau- sible solutions. Three main strategies emerge for introducing learning bias: the first relies on applying PINNs, as demonstrated across a broad range of assets, such as aero-engine spline cou- plings (A. Wang et al., 2025), power trans- formers (Ramirez et al., 2024), axial piston pumps (Chao et al., 2025; C. Dong et al., 2025; Wang, Z. et al., 2023), and lithium-ion batter- ies (Jang et al., 2025; Y. Liu et al., 2025; S. Singh et al., 2023; F. Wang, Zhai, et al., 2024; L. Wang, Yang, & Hu, 2025; Wen et al., 2023). The second strategy hinges on embedding mechanistic models as soft constraints in otherwise standard ML mod- els, including crack growth and wear laws (Badora et al., 2023; Y. Pan et al., 2025), half-cell or elec- trochemical aging models and equivalent-circuit dynamics (Navidi et al., 2024; Pugalenthi et al., 2024; Z. Xu et al., 2022; S. Zhang, Liu, Xu, Chen, & Su, 2025; S. Zhang, Liu, Xu, Guo, & Su, 2025), and dynamic models of industrial robots derived from joint multi-energy and rigid-body dynam- ics (R. Li et al., 2024; S. Wang et al., 2024). Lastly, a third set of strategies relies on com- paratively simple constraints, with the two most prominent patterns either imposing generic degra- dation properties such as monotonicity on health indicators or degradation trajectories (Y. Deng et al., 2025; Fassi et al., 2024a; Freeman et al., 2022; Najera-Flores et al., 2023; Y. Wang et al., 2024; X. Xu & Liu, 2024) or encoding prior knowledge about fault signatures in the frequency domain (Qiao et al., 2024; Tang et al., 2024; Z. Xu et al., 2024). Among these, the PINN formulation has become ade factostandard for learning-bias integration, offering a unified way to weakly enforce the governing dynamics of a system, 50 whether mechanical, electrochemical, or other- wise. Its abstract, problem-agnostic formulation makes it straightforward to implement across diverse domains, which has contributed to their widespread adoptionâparticularly in PHM. Yet this convenience introduces a characteristic chal- lenge: the relative weighting between data-fidelity and physics-residual losses. Across the reviewed literature, loss balancing is predominantly con- trolled by scalar weighting coefficients that are either fixed or tuned empirically, with only a few exceptions proposing more systematic schemes such as uncertainty-based weighting (Wen et al., 2023), Bayesian optimization-based weight- ing (F. Wang et al., 2025), or gradient-norm-based balancing (S. Singh et al., 2023; S. Zhang, Liu, Xu, Guo, & Su, 2025). When these weights are poorly calibrated, the model may overfit the data while failing to satisfy physical constraints, or vice versaâa tension that remains an open practical challenge. Notably, among all studies incorporating a learning bias, only a single study does so through the optimization procedure itself: Tefera et al. (2025) embed monotonicity, boundary, and energy-consistency constraints directly into the gradient-descent updates. Whether optimization- level integration of physics offers practical advan- tages over loss-based regularization (e.g., in settings with multiple competing constraints) remains an open question, given that only a sin- gle study has explored this pathway. However, the near-complete dominance of regularization-based approaches likely reflects practical considerations. Adding penalty terms is trivial in modern ML frameworks, whereas implementing custom opti- mizers demands specialized expertise and is harder to generalize. Ultimately, a key trade-off emerges within this class: excluding qualitative degradation properties (e.g., monotonicity or boundedness), introduc- ing learning bias inherently couples the PIML formulation to the specific asset, trading the methodâs transferability across different scenar- ios for stronger constraints within the intended domain. An example of a more transferable design is the Koopman-informed N by E. Wang et al. (2025), which learns an approximately lin- ear evolution in a latent eigenfunction space and regularizes forward-backward consistency of the dynamics without relying on asset-specific mech- anistic models. 5.1.4 Hybrid Approaches Hybrid approaches couple independent physics- based and ML models either in parallel or in series, without embedding physical knowledge within the ML pipeline itself. Across all four classes, hybrid approaches exhibit the most pronounced method- ological uniformity, arising from the limited struc- tural design space: one model feeds the other, or both run concurrently with combined outputs. The dominant pattern is in-series coupling in which physics informs ML. Several studies leverage calibrated physics-based models, namely ECMs (X. Feng et al., 2024; Lin et al., 2025), elec- trochemical and aging models for batteries (L. Ma et al., 2024; Shi et al., 2022; B. Sun et al., 2023; L. Xu et al., 2023), fuel-cell models (Pet- torossi et al., 2025), and wear and degradation laws (K. Zhu et al., 2024; Y. Zhu, Cheng, et al., 2024) to derive intermediate quantities (e.g., model parameters or health indicators) that serve as inputs to an ML model performing the actual PHM task. In effect, the physics-based models act asmechanism-alignedencoders that compress raw sensor data into compact, physically interpretable features. By reducing the input dimensionality to as few as six (Lin et al., 2025) or ten parame- ters (L. Ma et al., 2024), this in-series coupling enables simplifying the downstream learning task, justifying the use of simple models such as RFs and shallow fully connected networks. Moreover, in three other cases, the physics- based model runs alongside a data-driven encoder, with their outputs concatenated and fed to a sin- gle ML model. From the viewpoint of the ML models performing fault diagnosis (S.K. Singh et al., 2024), health assessment (F. Wang, Wu, et al., 2024), and RUL prediction (Liang et al., 2024), respectively, this configuration effec- tively retains an in-series coupling. This approach arises when the physics-based model is acknowl- edged as too low-fidelity to serve as the sole encoder (unlike the in-series approaches discussed above), yet still contributes structured informa- tion absent from raw data. In each case, the architecture implicitly decomposes the prediction task: the physics branch supplies trend-level or steady-state features while the data-driven branch 51 captures residual dynamics or high-frequency fluc- tuations. Ablation studies in two of the three studies confirm that the combined approach out- performs either branch in isolation, while the third (S.K. Singh et al., 2024) demonstrates clear performance gains over the data-driven branch alone. The reverse coupling (ML informing physics) is less common but mostly follows a consistent logic: the physics-based model requires as input a quantity that is not directly observable from available sensors, and the ML modelâs sole task is to estimate precisely this quantity. Identified examples include a GPR model that infers lifetime percentage to correct particle states in a Paris law-based crack-growth model (T. Li, 2024), an RFR that maps vibration features to the current pitting area for a Paris law-inspired pit-growth model (Kundu et al., 2024), a deep RL agent that updates flow correction coefficients of a first- principles system-level digital twin of a fuel control system (W. Xu, Wang, et al., 2024), and a Wasser- stein GAN that infers local fault evidence which is used to update node weights of the proba- bilistic fault root cause tracing network (J. Xu et al., 2023). Crucially, the physics-based model retains responsibility for the mechanistic inference that constitutes the actual PHM task, thereby preserving physically consistent predictions. In-parallel approaches are the least studied, with all identified examples addressing lithium- ion batteries, where data-driven residual mod- els run alongside electrochemical and equivalent- circuit battery models (Firoozi et al., 2022; Kohtz & Wang, 2022; L. Zhang et al., 2024). All three studies share a common residual-correction scheme: a physics-based model (equivalent-circuit or electrochemical) provides a structured base- line prediction, while the ML component (N, BiLSTM or GPR) explicitly learns the discrep- ancy between that prediction and the measured signal. The concentration of in-parallel approaches in battery applications likely reflects the availabil- ity of compact, well-understood physics models whose outputs are directly comparable to sensor measurementsâa prerequisite for meaningfully defining a learnable residual. Hybrid approaches offer clear practical advan- tages: they require comparatively low engineering effort provided a physics-based model of sufficient fidelity exists. However, in configurations where the physics model informs a downstream ML model, the latter remains a black box, inheriting the limitations of purely data-driven techniquesâ most notably the risk of physically implausible predictions and unreliable extrapolation. Nev- ertheless, hybrid approaches are still relatively common in the PHM literature (see Fig. 4). 5.1.5 Multi-Class Approaches Although the four classes are conceptually dis- tinct, the literature confirms that they are comple- mentary in practice. Although studies have been assigned to multiple classes only if their mecha- nisms for incorporation are clearly separable, five works nevertheless span multiple classes (Badora et al., 2023; S. Li et al., 2025; H. Sun et al., 2022; F. Wang, Wu, et al., 2024; Y. Zhu, Cheng, et al., 2024), and in all cases a learning-bias component is present. This pattern likely reflects the comparatively low implementation barrier of adding physics-informed regularization, which naturally facilitates its combination with the other approaches. However, none of these multi-class studies report ablation experiments that isolate the contribution of each embedded mechanism. To conclude, the multi-class cases should pri- marily be interpreted as evidence that the four classes are practically compatible, rather than as proof that specific combinations outperform care- fully designed single-class approaches. Establish- ing when and how multi-mechanism designs offer systematic advantages remains an open question. 5.2 The Effects of Incorporating Physics Having characterized the methodological land- scape in Section 5.1, this section examines the empirical evidence for the effects of incorporat- ing prior physical knowledge into ML-based PHM. The analysis is organized along six dimensions: predictive performance, generalizability, robust- ness, data efficiency, interpretability, and physical consistency. For each dimension, the strength and scope of available evidence is assessed, andâwhere possibleâthe observed effects are traced back to the class of incorporation. The section concludes by identifying systematic gaps and biases in how effects are currently reported. 52 5.2.1 Predictive Performance Improved predictive performance constitutes the most consistently reported benefit and is sup- ported across all four classes of approaches and all four PHM tasks. For fault detection and diagnosis, numerous studies report higher classification accu- racy relative to baselines (Y. Dong et al., 2022; C. Gao et al., 2024; Y. Huang et al., 2025; Z. Ma et al., 2025; Qiao et al., 2024; Z. Xu et al., 2024; Zeng et al., 2025; Y. Zhu, Zi, et al., 2024). For health assessment and prognosis, reduced regres- sion errors are typical (Mei et al., 2024; Najera- Flores et al., 2023; L. Wang, Yang, & Hu, 2025; Yucesan & Viana, 2022). While this breadth of evidence is encouraging, its interpretation requires caution for two reasons. First, the strength of the evidence varies sub- stantially with experimental design. A minority of studies include clear ablation experiments com- paring the proposed model against an architec- turally identical counterpart from which only the physics component has been removed. The major- ity instead compare against simple baselines (e.g., standalone SVR, RF, or LSTM), even when the proposed PIML model is architecturally far more complex. Under such conditions, it is difficult to disentangle performance gains attributable to the embedded physics from those arising from increased model capacity, additional engineering effort, or more sophisticated training procedures. Comparisons against strong, state-of-the-art base- lines are rare (S. Liu et al., 2025; E. Wang et al., 2025), further limiting the conclusiveness of reported performance gains. Second, the magnitude of reported improve- ments is almost never contextualized with respect to practical significance. In PHM, a small reduc- tion in the RUL prediction error may or may not alter maintenance decisions depending on the assetâs failure consequences, the planning horizon, and the associated prediction uncertainty. Yet no study in the reviewed literature connects reported accuracy gains to downstream decision quality or maintenance cost savings. This omission lim- its the practical value of reported performance improvements for deployment contexts. Moreover, the prevalence of highly problem- specific solutions precludes drawing overarching conclusions about which type of prior knowl- edge or integration pathway yields the greatest performance gains in general. 5.2.2 Generalizability Generalization, the ability to maintain perfor- mance under distributional shift, is a less fre- quently but repeatedly reported benefit. Evidence emerges predominantly from two experimental protocols: either in TL settings (F. Wang, Zhai, et al., 2024; Y. Zhang, Feng, et al., 2023; Y. Zhang et al., 2024; Y. Zhu, Zi, et al., 2024), or across differ- ent operating and environmental conditions (such as varying loads, speeds, or temperatures) (Abiria et al., 2025; Y. Feng et al., 2022; E. Wang et al., 2025; Zeng et al., 2025; K. Zhu et al., 2024). A critical distinction, seldom made explicit in the reviewed literature, is that between in- distribution and out-of-distribution generaliza- tion. The former refers to conditions spanned by the training data but held out for evaluation, whereas the latter refers to genuinely novel con- ditions, environments, or assets. Most reported generalization evidence pertains to the former. True out-of-distribution generalization, which rep- resents the stronger and practically more rele- vant claim, is almost never targeted directly and remains largely unsubstantiated. Observational-bias approaches are particularly well-represented among generalization claims. However, their purported generalization advan- tage must be interpreted carefully: by augmenting the training distribution with simulated data that spans a broader operating regime, the effective training distribution is expanded. Improvements relative to a data-driven baseline trained on less data therefore partly reflect the additional infor- mation injected via simulation, rather than a structural capacity to generalize. Without con- trolling for training-set coverage, such claims risk conflating data augmentation with genuine gener- alization ability. 5.2.3 Robustness Robustness encompasses a broad range of notions in the ML literature, including stability under input perturbations, resilience to label noise, and resistance to distributional shift. C. Yin et al. (2025), Zeng et al. (2025), and S.K. Singh et al. (2024), for example, investigate robustness 53 against noisy data. Bajarunas et al. (2024) evalu- ate robustness to distributional shift by comparing HI quality and RUL prediction performance across methods. Other studies interpret robustness dif- ferently, including low variance in RUL prediction errors across varying battery chemistries and oper- ating conditions (L. Ma et al., 2024) and the modelâs ability to avoid false alarms (Jin et al., 2024). In the latter case, however, evaluation was conducted on a single turbine, with only one doc- umented true fault and a few false-alarm episodes, without broader tests across turbines, fault types, or operating conditionsâlimiting the generaliz- ability of the finding. The inconsistent opera- tionalization of robustness across studies pre- cludes aggregating evidence into a coherent assess- ment. While PIML shows promise in this dimen- sion, results are fragmented and often context- specific, leaving the overall evidence inconclusive. 5.2.4 Data Efficiency Data efficiency, the ability to achieve a given performance level with fewer real observations, is rarely evaluated through dedicated experimen- tal protocols. Where evidence exists, it typically derives from iteratively reducing dataset sizes and observing performance degradation (Nguyen et al., 2023; Tang et al., 2024; F. Wang, Zhai, et al., 2024; S. Zhang, Liu, Xu, Guo, & Su, 2025), or from limiting the available history or opera- tional horizon (E et al., 2025; Fu et al., 2024; Liang et al., 2024; K. Zhu et al., 2024). The available results generally suggest that incorporat- ing physics largely preserves performance as data volume decreases. Improved extrapolation capability can be regarded as a manifestation of data efficiency: by imposing physical constraints, a model can accu- rately recover solutions with fewer observations in poorly explored regions of the input space, as demonstrated by Z. Xu et al. (2022), S. Singh et al. (2023), and S.K. Singh et al. (2024). How- ever, the same qualification noted above applies to observational-bias methods: the claim of requir- ing little real data is only partially justified when the underlying data is effectively augmented with simulated observations. The actual data budget (including simulation data generation, physics- model calibration, and real data collection) is rarely reported transparently, obscuring the true resource requirements. 5.2.5 Interpretability Improved interpretability is among the most fre- quently claimed yet least substantiated benefits. In several instances, studies assert interpretability solely because prior physical knowledge has been incorporated, without providing any empirical evi- dence. Incorporating prior knowledge does not inherently make a model interpretable. Indeed, many PIML models remain functional black boxes that offer negligible insight into how predictions are derived. A few exceptions demonstrate that at least parts of the modelâs inner workings may exhibit a degree of interpretability, as shown by Z. Xu et al. (2024), where internal representations qualitatively align with known fault frequencies and uncertainty patterns. Several studies illus- trate how integrating prior physical knowledge enhances the separability of learned representa- tions between regimes, classes, or operating con- ditions, often visualizing this effect using methods such as t-distributed stochastic neighbor embed- ding (Cheng et al., 2024; Y. Dong et al., 2022; Y. Huang et al., 2025; S. Li et al., 2024; S. Liu et al., 2025; Zeng et al., 2025; Y. Zhu, Zi, et al., 2024). While such observations suggest that physics guides the model toward more structured feature spaces, improved cluster separability does not constitute interpretability in a rigorous sense. Consequently, interpretability remains an aspi- ration rather than a demonstrated outcome of current PIML methods in PHM. 5.2.6 Physical Consistency Physical consistency is frequently claimed but lacks a shared definition, rendering it context- dependent. As used in the reviewed literature, it encompasses: (i) adherence to fundamental phys- ical laws (e.g., conservation principles, thermody- namics, or electrochemistry); (i) conformity with established degradation properties (e.g., mono- tonicity, irreversibility, or bounded ranges); and (i) admissibility of internal model variables (e.g., SOCâ[0,1], crack lengthâĽ0, or temperature within certain limits). As established earlier, the four classes offer fundamentally different struc- tural guarantees that must be considered when 54 formulating or interpreting physical consistency claims (see Sec. 5.1). Beyond these class-level differences, several structural limitations deserve emphasis. First, many constraints enforce local behavior (e.g., step-to-step monotonicity), without guaranteeing globally realistic trajectories (e.g., correct knee behavior in battery aging). Second, PINNs enforce governing equations only at a finite set of collo- cation points. Third, constraints typically cover only a subset of the physics (e.g., simple wear law), while ignoring multi-physics coupling effects that may dominate in certain operating regimes. Consequently, partial enforcement of physical con- straints can create a misleading impression of physically consistent behavior when the unmod- eled physics becomes dominant. Ultimately, the field lacks a common metric for quantifying the degree of physical consistency of a model, which in turn presupposes consensus on what this notion precisely entails. Without such metrics, the claims of physical consistency remain qualitative and largely unverifiable. 5.2.7 Caveats on Reported Effects Beyond the dimension-specific observations above, several cross-cutting issues affect the reliability of the overall evidence base. First, the absence of ablation experiments in many studies pre- vents isolating the contribution of incorporated physics from confounding factors such as architec- ture changes, additional hyperparameter tuning, or increased training data. This is particularly acute in studies spanning multiple classes (Badora et al., 2023; S. Li et al., 2025; H. Sun et al., 2022; F. Wang, Wu, et al., 2024; Y. Zhu, Cheng, et al., 2024), where no ablation experiment disentangles the individual mechanisms. Second, PIML introduces a unique and under- recognized risk of evaluation leakage: when physics-model parameters are calibrated on data that overlaps with the test setâas in Abiria et al. (2025), where Basquinâs and Parisâ law parameters are fitted to the complete dataset before train- test splittingâthe physics prior becomes partially informed by the test data itself. This system- atically favors the physics-informed model over purely data-driven baselines and renders general- ization claims overly optimistic. The vulnerability extends beyond this single instance: any PIML approach that calibrates embedded physics-model parameters from data is susceptible to this form of leakage unless the calibration is strictly confined to the training partition. Third, potential drawbacks of incorporating physics are almost never quantified. Computa- tional cost (during training and inference), conver- gence behavior, sensitivity to loss-weight selection, and implementation overhead relative to purely data-driven alternatives are consistently omitted from evaluations. Among the reviewed studies, only S. Liu et al. (2025) report Floating Point Operations (FLOPs) for all compared models. The widespread absence of such information represents a significant barrier to informed method selection and yields a strongly benefit-skewed evidence base that is insufficient as a foundation for balanced deployment decisions. 5.3 Contextual Limitations Having discussed the effects of incorporating physics into ML in Section 5.2, this section high- lights the contextual limitations that must be considered when assessing the maturity and gen- eralizability of PIML in PHM. It first documents a pronounced concentration of the reviewed liter- ature around a narrow set of assets, then traces this imbalance to two reinforcing drivers, namely data availability and the availability of formalized prior physical knowledge, and finally discusses the resulting implications for maturity assessment. Across the reviewed literature, lithium-ion bat- teries and bearings clearly emerge as the primary focus, accounting for approximately half of the total studies (see Fig. 4). Only a small subset of other assets (i.e., cutting tools, pumps, tur- bofan engines, and metal specimens) has been investigated in five or more studies, highlighting their relatively limited attention. The remain- ing assets are examined sporadically, often in a single study, underscoring a significant gap in research coverage. This concentration likely reflects the convergence of multiple factors: the industrial and commercial significance of batter- ies and bearings, the maturity of their respective research communities, and (as discussed below) the favorable availability of both public datasets and well-characterized physics-based models for these assets. These factors are mutually reinforc- ing rather than independent. 55 ML research is strongly shaped by data avail- ability, which in turn influences both the problems studied and the methods developed. The work of Mauthe, Steinmann, Neu, and Zeiler (2025) presents the most comprehensive overview and analysis of publicly available degradation datasets for PHM, covering 98 datasets in total. The ongo- ing updating of this overview, along with com- plete documentation for each dataset, is available online (Mauthe, Braun, Raible, Zeiler, & Huber, 2024). Notably, batteries and bearings form the two largest asset categories in terms of dataset count, with 15 each. By contrast, most other asset types are represented by only one or two datasets. This imbalance means that researchers reliant on publicly available data encounter a markedly richer landscape for batteries and bear- ings than for other assets. In addition, specific datasets (such as the battery dataset provided by the NASA Prognostics Center of Excellence (Saha & Goebel, 2007), XJTU-SY (B. Wang, Lei, Li, & Li, 2020) and FEMTO (Nectoux et al., 2012) for bearings, or C-MAPSS for turbofan engines (Sax- ena & Goebel, 2008)) have acquired the status of de factocommunity benchmarks, further concen- trating research activity around the assets they represent. The reviewed literature reflects this pattern, drawing heavily on publicly available datasets and exhibiting a similar asset distribu- tion. Consequently, the predominance of studies on lithium-ion batteries and bearings is, at least in part, reinforced by an availability bias. The review reveals a dependency between the degree of formalization of available prior phys- ical knowledge and the range of incorporation strategies that become feasible. Lithium-ion bat- teries and bearings are particularly amenable to PIML because their underlying physics is well-characterized and formalized. For batter- ies, electrochemical modelsâincluding SPM vari- ants (Y. Liu et al., 2024; S. Singh et al., 2023; S. Zhang, Liu, Xu, Chen, & Su, 2025; S. Zhang, Liu, Xu, Guo, & Su, 2025), half-cell (Navidi et al., 2024; Thelen et al., 2022) and SEI-growth mod- els (Kohtz & Wang, 2022; Y. Liu et al., 2025)âare widely employed, alongside ECMs for cell voltage and SOC dynamics (Fu et al., 2024; X. Liu et al., 2024; L. Qin et al., 2025). For bearings, multi-DOF dynamic models (Y. Deng et al., 2023; Y. Dong et al., 2022; Y. Qin et al., 2024, 2025; S. Sun et al., 2024; Y. Zhang, Feng, et al., 2023), analyti- cal vibration signal models (F. Gao, Zhang, You, & Cao, 2024; Y. Li et al., 2024; Y. Zhu, Cheng, et al., 2024), and fault characteristic frequencies (C. Gao et al., 2024; Qiao et al., 2024; Z. Xu et al., 2024; Zeng et al., 2025) provide a rich repository of embeddable prior knowledge. Accordingly, the lit- erature appears to be influenced, at least in part, by a methodological selection bias. This creates a self-reinforcing pattern in which assets whose physics is already well-formalized attract dispro- portionate research attention, while those whose degradation involves poorly understood or multi- physics mechanisms remain underrepresented. These contextual limitations must inform any evaluation of the current maturity of PIML in PHM. While the mere volume of studies covered in this review may suggest that PIML has matured into an established standard for industrial PHM, such an interpretation would be misleading. The observed imbalance necessitates a more differen- tiated assessment. For lithium-ion batteries and bearings, a certain level of methodological matu- rity can reasonably be claimed. However, even for these assets, existing studies focus predominantly on specific PHM tasks: health assessment and prognosis for batteries; diagnosis and prognosis for bearings. Thus, the apparent maturity is con- fined to a few asset-task combinations rather than the full PHM spectrum. Beyond these focal assets, PIML research remains at an early, exploratory stage with only scattered and often isolated evi- dence. The skew in asset coverage is particularly consequential because PIML methods are gener- ally tied to the specific asset under study, owing to the use case-specific nature of the embedded prior knowledge. This entanglement constrains applicability to the studied context, and, by exten- sion, raises the question of how to leverage PIML without sacrificing generalizability. 5.4 Barriers to Deployment The concentration of the literature around a nar- row set of assets and tasks (see Sec. 5.3) already constitutes a barrier to the broader deployment of PIML-based PHM. Even within these well-studied domains, however, a substantial gap separates current proofs of concept from industrially deploy- able solutions. This section examines the practical barriers that collectively account for this gap. 56 These barriers are not independent but cumu- lative: constructing a PIML solution demands significant expertise and engineering effort; even where such effort is invested, the resulting models rarely provide decision-relevant uncertainty esti- mates; and even if both of the former barriers were overcome, insufficient evidence exists regarding whether these models can operate under real-time and resource constraints. 5.4.1 Implementation Effort and Expertise A prerequisite for any PIML solution is the suc- cessful identification, formalization, and embed- ding of appropriate prior physical knowledgeâa process that remains largely undocumented and unquantified across the reviewed literature. Unlike purely data-driven pipelines, which can often be constructed by ML practitioners with general domain familiarity, PIML demands expertise at the intersection of two traditionally separate dis- ciplines: the physics of the assetâs degradation mechanisms and the engineering of ML architec- tures and training procedures. This dual require- ment manifests at multiple stages: selecting which physics to embed (and, equally importantly, which to omit); translating qualitative physical under- standing into a formal, computable representation amenable to integration; choosing an appropri- ate pathway (observational, inductive, or learning bias); and calibrating the interplay between data- driven and physics-informed components, such as loss-weight tuning or simulator fidelity. No study among those reviewed reports the human effort, development time, or iterative design cycles required to arrive at the final physics-informed model. Yet this engineering over- head is arguably the most immediate practical barrier, particularly when seeking to deploy PIML at scale across heterogeneous asset fleets. The observation from Sections 5.1 and 5.3 that nearly all solutions are tightly coupled to the specific asset under study is, in part, a downstream conse- quence of this barrier: each new asset demands the aforementioned integration effort, which cannot be easily amortized. Incorporating more general prior knowledge, such as simple monotonicity constraints, can broaden applicability, although the resulting per- formance gains may often be modest. Developing methods that are both applicable across diverse contexts (or even assets) and capable of deliver- ing substantial improvements thus remains a key challenge. Nevertheless, three patterns in the lit- erature suggest pathways toward reducing this overhead by circumventing the need for intricate physical models: (i) structural and topological knowledge describing an assetâs component layout or the relative placement of sensors is particu- larly suitable for graph-based approaches (Cheng et al., 2024; Y. Feng et al., 2022; Jin et al., 2024; S. Liu et al., 2025; M. Zhou et al., 2025); (i) qualitative degradation properties (e.g., mono- tonicity) require no system-specific physical model and can be imposed via architectural constraints, such as monotone activations, constrained hidden- state updates, and bounded output layers (Abiria et al., 2025; Bai et al., 2023; Hao et al., 2023; S. Li et al., 2025; C. Yin et al., 2025; Z. Zhou et al., 2023), or inequality-type loss terms that penalize local violations (Y. Deng et al., 2025; Fassi et al., 2024a; Najera-Flores et al., 2023; F. Wang, Zhai, et al., 2024); and (i) causal relationships between operating conditions, sen- sor signals, and the underlying degradation state, although studied less frequently (Bajarunas et al., 2024). Beyond this, very few studies actu- ally demonstrate applicability across multiple use cases without requiring modifications (Bajarunas et al., 2024; Tang et al., 2024; E. Wang et al., 2025; Y. Wang et al., 2024; Z. Zhou et al., 2023), and in each case separate training is still required. Taken together, these structural, qualitative, and causal forms of prior knowledge represent the most accessible entry points for transferable PIML in PHM. However, transferable PIML solutions remain an aspiration rather than an established practice, with the high implementation cost per asset being a principal inhibitor. 5.4.2 Uncertainty Quantification Uncertainty quantification is widely recognized as a core requirement for prognostics in PHM, where single-point estimates are âusually considered meaninglessâ for industrial applications (Kundu et al., 2020). Yet the vast majority of reviewed stud- ies produce exclusively deterministic predictions, offering no calibrated confidence information to inform decision-making. 57 A smaller subset of studies combines physics with probabilistic models (Bai et al., 2023; Ellis et al., 2022; C. Jiang et al., 2025). In addi- tion, Bayesian NNs are used, though to a lesser extent (Y. Deng et al., 2023; Liang et al., 2024; Najera-Flores et al., 2023). In another study, a Wiener process-based stochastic degradation model is embedded in an N (Z. He et al., 2025). However, among these, very few explicitly inves- tigate how the incorporation of prior physical knowledge improves the quality of uncertainty estimates relative to purely data-driven proba- bilistic models (Bai et al., 2023; Z. Xu et al., 2024). This represents a missed opportunity, because physics-informed constraints (including inductive and learning bias) have the potential to sharpen predictive distributions, e.g., by ruling out phys- ically implausible predictions and thus yielding tighter confidence bounds. Conversely, this same mechanism introduces a distinctive risk: in operating regimes where the embedded physics becomes inaccurate (e.g., due to unmodeled multi-physics coupling, degradation- mode transitions, or environmental conditions outside the modelâs validity range), overly con- strained physics-informed predictions may pro- duce dangerously miscalibrated uncertainty esti- mates. However, neither improvements nor dete- riorations in uncertainty estimates resulting from the incorporation of prior physical knowledge have been sufficiently studied in the current literature. As a result, PIML models in PHM rarely provide the reliable, calibrated uncertainty information needed for decision-making, representing a critical barrier to deployment in settings where decisions carry safety or financial consequences. 5.4.3 Computational Cost and Operational Readiness The transition from offline validation to opera- tional deployment introduces requirements that the current literature leaves largely unaddressed: real-time inference under latency constraints, exe- cution on resource-limited hardware, adaptation to evolving conditions, and integration with exist- ing monitoring and maintenance infrastructure. Assessing the feasibility of meeting these require- ments presupposes insight into computational costs, yet such information is largely absent from the reviewed studies. Among all reviewed studies, only S. Liu et al. (2025) report FLOPs for all compared models, thereby enabling fully transparent computational assessment. Partial reporting is more common but insufficient: Fu et al. (2024) and F. Xie et al. (2024) report FLOPs for their proposed physics- informed models but not for baselines. Zeng et al. (2025) and Pugalenthi et al. (2024) evaluate efficiency solely in terms of training time with- out reporting FLOPs or parameter counts, lim- iting the comparability of their efficiency claims across studies. L. Wang, Yang, and Hu (2025) additionally report inference times, revealing that their PINN-based approach incurs training times approximately two orders of magnitude above the fastest baseline, while inference times remain comparable across models. These scattered obser- vations do not enable systematic comparison of performance gains versus computational overhead across the field. Nevertheless, the inherent characteristics of each class of approaches permit a qualita- tive assessment of computational trade-offs. Observational-bias approaches shift the computa- tional burden to an offline simulation phase: once the ML model is trained, inference cost is identi- cal to a purely data-driven model. Inductive-bias approaches embed physics directly in the archi- tecture, where the added inference cost depends on the specific mechanismâranging from neg- ligible (e.g., constrained activations) to non- trivial (e.g., embedded ODE solvers). Learning- bias approaches, particularly PINNs, increase training cost through additional loss evaluations and automatic differentiation but typically incur no overhead at inference, unless online retrain- ing is required. Hybrid approaches are unique in retaining the physics-based model during infer- ence, which can become a bottleneck for real-time deployment when the physics model is computa- tionally expensive. These qualitative distinctions provide initial guidance for practitioners facing real-time engineering decisions, while underscor- ing the urgent need for future studies to system- atically report computational metrics alongside predictive performance. Beyond computational cost in isolation, sev- eral interrelated deployment requirements remain entirely unaddressed. First, virtually no study reports inference latency, throughput, or mem- ory footprint under conditions representative of 58 industrial monitoring systems. Second, experi- ments investigating whether current PIML models can be executed on resource-constrained hardware such as embedded edge devices are missing. Third, the coupling of PIML models with operational infrastructure (e.g., programmable logic con- trollers, supervisory control and data acquisition systems, cloud- and edge-based data pipelines, or maintenance management systems) is never discussed. Fourth, the question of model main- tenance after deployment, including adaptation when operating conditions drift, physics assump- tions degrade, or new failure modes emerge, is entirely unstudied. Based on the experimental settings reported (i.e., predominantly laboratory datasets, offline evaluations, and controlled condi- tions), the reviewed methods appear to correspond broadly to early Technology Readiness Levels (TRLs) (approximately TRL 3â4), with no study demonstrating operational deployment. 5.5 Terminology The review highlights a notable inconsistency in the terminology regarding PIML across PHM studies. Within this paradigm, many researchers introduce novel contributions by combining the descriptor âphysics-informedâ with a term denot- ing their specific model or approach. As a result, the descriptor âphysics-informedâ emerges as the most frequently used term among the reviewed studies, effectively signaling that prior physical knowledge is incorporated into the ML pipeline in some form. This dominance, how- ever, may have prompted some researchers to use other descriptors better reflecting the par- ticular characteristics of their approach, such as âphysics-guidedâ (S. Li et al., 2025) or âphysics- constrainedâ (Najera-Flores et al., 2023)âa phe- nomenon known to the authors prior to con- ducting this review, which also influenced the keyword selection (see Tab. 2). While the choice of the respective descriptor may be appropriate in some instances, it is rarely accompanied by an explicit explanation. The wide variety of terms used is, on the one hand, likely a consequence of the relatively recent emergence of PIML as a research field, particularly in older studies. On the other hand, in certain cases, the descriptor appears to reflect an arbitrary choice of word- ing. In milder instances, this results in descrip- tors other than âphysics-informedâ being used consistently within a single studyâindividually unproblematic, yet collectively contributing to terminological fragmentation across the field. In more extreme cases, however, multiple variants may appear within the same study. For instance, Freeman et al. (2022) refer to their loss func- tion as âphysics-informed,â âphysics-guided,â and âphysics-based,â which likely reflects an attempt to employ synonyms for stylistic variation. While inconsistent terminology across studies is under- standable, given that the seminal work by Kar- niadakis et al. (2021) was published in 2021, inconsistencies within individual studies remain problematic, as they can obscure the intended message and compromise clarity for the reader. When the descriptor âphysics-informedâ is used consistently, an ambiguity naturally arises with the term PINNâa pattern repeatedly observed across the reviewed studies. Since Raissi, Perdikaris, and Karniadakis (2019) coined the term to describe NNs that incorporate known physical laws, expressed as differential equations, into their loss function, the term PINN is some- what constrained in its usage. Several studies make use of this term, yet their approaches do not align with its original definition (Badora et al., 2023; Y. Deng et al., 2025; Navidi et al., 2023). In light of the highly influential work of Raissi et al. (2019), many readers may reasonably expect the term PINN to imply the incorpora- tion of differential equations for regularization, creating potential confusion when used differently. It is acknowledged, however, that establishing clear terminology for novel approaches that are both physics-informed and employ NNs, yet do not align with the concept of PINNs, remains a challenge. Since approaches to implementing PHM are typically subdivided into model-based, data- driven, or hybrid, it is understandable that some studies use the terms âphysics-informedâ and âhybridâ interchangeably (e.g., Lehmann et al. (2024)), given that the latter is generally regarded as a broader category encompassing PIML. While this may be technically correct, the classification scheme adopted in this review makes an explicit distinction between PIML and hybrid approaches to further enhance clarity in this regard, thereby 59 facilitating more precise communication regarding the methodology employed in each study. In this context, the combined use of these terms within a single phrase can be misleading, such as âhybrid physics-informed neural networkâ (Dourado & Viana, 2022), âphysics-informed spatio-temporal hybrid neural networkâ (M. Zhou et al., 2025), or âhybrid physics-embedded recurrent neural net- workâ (R. Li et al., 2024). This does not pose a problem when the corresponding study actually combines a hybrid approach that incorporates bias via one of the three PIML pathways. Apart from the use of descriptors to spec- ify the proposed approach, some studies even apply them to the entire field of research, disre- garding the foundational definition of PIML by Karniadakis et al. (2021). While these alterna- tive terms still refer to research that effectively incorporates prior physical knowledge into ML, this practice appears to be related to the earlier- discussed issue of unnecessary lexical variation. Although there is some flexibility in naming a novel approach differently, using alternative terms for the research field may give the impression of a subtle yet notable distinction regarding PIML. For example, Kohtz and Wang (2022) propose employing âphysics-based machine learning tech- niques.â Bai et al. (2023) deviate even further, referring to it as âknowledge-constrained machine learning.â R. Yan et al. (2025) claim to intro- duce a novel taxonomy, named âknowledge driven machine learning,â intended to synthesize the cur- rent state of research at the intersection of prior knowledge and ML. They explicitly adopt the taxonomy of von Rueden et al. (2021), yet con- tribute no novelty and could therefore have been referred to simply as IML. C. Yin et al. (2025) reference Karniadakis et al. (2021) to outline the three pathways for introducing physics into ML. In doing so, they fully adopt the conceptualization of PIML, yet deliberately use the term âphysics- guidedâ throughout their entire studyâa choice for which no clear rationale is provided. 5.6 Future Research The preceding analysis reveals that PIML already delivers tangible performance benefits for PHM, yet persistent methodological gaps (see Sec. 5.1), insufficiently substantiated claims (see Sec. 5.2), narrow asset coverage (see Sec. 5.3), practical deployment barriers (see Sec. 5.4), and termino- logical inconsistencies (see Sec. 5.5) collectively constrain the fieldâs advancement. The following directions are organized to address each of these gaps in turn. The review demonstrates the potential of PIML for PHM, yet the analyzed studies also reveal persistent methodological gaps and open challenges, as discussed in Section 5.1. Addressing these issues will be essential to translate cur- rent approaches into robust, deployable solutions for industrial practice. Building on the synthe- sis and critical discussion presented above, one key opportunity for future research is the design of benchmark studies that systematically com- pare observational-, inductive-, and learning-bias approaches under controlled conditions. Wherever feasible, these benchmarks ought to reuse the same prior physical knowledge so that differences in performance can be attributed to the pathway of integration rather than to the physics itself. The resulting evidence would enable well-founded guidelines for selecting an integration strategy based on the type and fidelity of available prior knowledge, the volume and quality of data, and application-level requirements such as robustness and interpretability. Tackling the highly heterogeneous landscape of inductive-bias approaches (see Sec. 5.1.2) requires moving from problem-specific designs toward modular building blocks that can be reused with minimal adaptationâanalogous to the PINN framework for learning-bias approaches. Ultimately, the goal should be to reduce the often-overlooked engineering overhead of PIML, making inductive-bias approaches more practi- cal and scalable. Moreover, regularization-based methods dominate the landscape of learning- bias approaches (see Sec. 5.1.3), yet reliance on empirically tuned loss balancing suggests a shift toward adaptive methods. While aiming to sta- bilize training and promote convergence, such methods should ultimately ensure that the phys- ical loss contributes appropriately, preventing the model from over-prioritizing the minimization of data loss. To gain more precise control over both training dynamics and the influence of physics on model training, future research should addi- tionally explore integrating physics directly into the optimizer. Although challenging to implement, physics-informed optimizers can produce solutions 60 that are both accurate and physically plausible by incorporating constraints directly into the opti- mization step, providing stronger adherence than loss-based regularization. While promising, the claimed improvements from incorporating prior physical knowledge largely remain unsubstantiated beyond predictive performance, and require more rigorous experi- mental design (see Sec. 5.2). To enable fair and informative comparisons, models built on sim- pler architectures should be benchmarked against conventional baselines, whereas more powerful architectures should be compared to state-of-the- art alternatives. In all cases, ablation studies are essential to isolate the specific contribution of the embedded physics. Moreover, experiments need to encompass diverse operating and environmen- tal conditions to rigorously evaluate the validity of the claimsâan aspect particularly crucial in the context of industrial PHM. In addition, phys- ical consistency often lacks a precise formulation. To facilitate meaningful comparisons, it is neces- sary to establish a formal definition and derive corresponding metrics. These could quantify the frequency and magnitude of constraint violations, deviations from physically plausible ranges, and cumulative errors over time, enabling a nuanced assessment of how well models respect physi- cal laws while maintaining predictive accuracy. Finally, to counter the prevailing benefit-skewed view, evaluations should routinely quantify the trade-offs and costs of PIML (e.g., computa- tional load, convergence behavior, training sta- bility, and engineering effort) alongside any per- formance gains. This would facilitate a balanced assessment of whether physics integration justi- fies its associated overhead in a given deployment context. Although deeper study of different asset types may mitigate the contextual limitations identified in Section 5.3, prioritizing generalizable solutions over asset-specific studies represents a more pro- ductive allocation of research effort. By systemat- ically analyzing and abstracting the mechanisms that have proven most effective, researchers can derive models applicable across entire classes of assets. This challenge hinges on shared fundamen- tal principles governing diverse degradation phe- nomena, highlighting the need to study trade-offs between asset-specific and general prior knowl- edge to understand how performance is gained or sacrificed in pursuit of broader applicability. A key enabler in this regard is modular design patterns that facilitate adaptation to new assets without requiring substantial engineering over- head. Taken to its logical conclusion, this points to the development of physics-informed foundational degradation models that are trained in a multi- task fashion to simultaneously address all core PHM tasks across multiple assets and operating conditions. Such models would essentially func- tion as universal backbones, obviating laborious problem-specific development. Translating PIML from research to indus- try faces the deployment barriers discussed in Section 5.4, necessitating progress along multi- ple interrelated axes. Complementing the efforts toward transferable solutions outlined above, automated or semi-automated methods for select- ing, calibrating, and embedding prior physical knowledge would lower the entry barrier for prac- titioners who possess domain expertise but lack specialized ML engineering skills. Moreover, uncertainty quantification must shift from an optional addition to an integral design objective. Future work should both sys- tematically investigate how incorporated physics affects the quality of uncertainty estimates and develop diagnostic indicators that signal when a modelâs physics assumptions are being violated, providing actionable safeguards against overconfi- dent predictions in safety-critical scenarios. Beyond this, future reporting should adopt full transparency regarding computational costs, while research should concurrently investigate lightweight physics-informed architectures suit- able for execution on resource-constrained edge hardware, including model compression and prun- ing techniques. Moreover, while generalization from simula- tion data to test bench data is frequently studied, the subsequent transfer to field data remains unaddressed. Future studies must demonstrate generalization to customized industrial machines, especially in settings where idealized physical laws may no longer apply because environmental influ- ences or multi-component interactions are super- imposed on the embedded prior knowledge. By extension, models must be capable of updating as operating conditions drift or new failure modes emerge, pointing to research on online adaptation mechanisms. 61 Finally, demonstrating closed-loop integra- tion with industrial monitoring and mainte- nance infrastructure (e.g., programmable logic controllers, supervisory control and data acquisi- tion systems, or cloud-edge data pipelines) would constitute a critical step toward elevating PIML- based PHM from laboratory validation to opera- tional maturity. In future research, reaching a consensus on core terminology would facilitate clearer commu- nication, more consistent methodology, and more meaningful synthesis of findings across studies. Addressing this challenge (identified in Sec. 5.5) entails both conceptual and practical efforts: con- ceptually, by establishing precise definitionsâsuch as that of physical consistencyâwhich can under- pin the development of relevant metrics, and practically, by adhering to a consistent naming convention when introducing novel methods. With respect to the latter, the results suggest adher- ing to âphysics-informedâ as the descriptor, in line with PIML. If deviation from this term is justified, either an explicit explanation should be provided or the alternative term should be unambiguous by default, as exemplified by the âKoopman-informed neural networkâ (E. Wang et al., 2025). In both cases, it is crucial that the chosen term be used consistently throughout the study. 6 Conclusion PHM is increasingly expected to provide reliable, trustworthy, and data-efficient decision support from sparse, noisy, and heterogeneous data. Given these demands, this review set out to examine how PIML is currently leveraged in PHM by sys- tematically addressing several research questions that explore the prior knowledge employed, its incorporation, and corresponding implications for practice. By conducting the most comprehensive systematic literature review to date at the inter- section of PIML and PHM, covering 212 studies, this work provides comprehensive answers to these questions. 1.Knowledge(a) What types of prior physical knowledge are being leveraged? In terms of prior physical knowledge, the field relies predominantly on mechanistic models and explicit degradation laws, whereas structural and causal relationships and qualitative degradation properties see less frequent application. 1.Knowledge(b) What forms of representa- tion are employed? Consequently, the corresponding forms of rep- resentation span from highly formalized expres- sions, through empirical and phenomenological formulations, down to implicit forms that capture principled assumptions. While the current focus capitalizes on well-established physical under- standing, evidence from the reviewed literature indicates that prior knowledge across the full spectrum of types and representations can be productively leveraged, with each type contribut- ing differently to the trade-off between physical fidelity and transferability. 2.Incorporation(a) How can prior physical knowledge be incorporated? Methodologically, three overarching pathways constitute distinct yet complementary classes of approaches to incorporating prior physical knowl- edge: observational bias, inductive bias, and learn- ing bias. Among these, learning-bias approaches are the most widely adopted, followed by inductive-bias and observational-bias approaches. This pattern likely reflects the advantages of learning-bias approaches in balancing flexibility regarding the types of prior physical knowledge that can be incorporated with practical imple- mentation feasibility, while still imposing soft con- straints that effectively regularize model behavior. In contrast, observational-bias approaches require a simulator of sufficient fidelity, whereas inductive- bias approaches face inherent difficulties in their tailored implementationâboth presenting practi- cal challenges that limit, to some extent, their adoption. Intermsofmethodologicalmaturity, observational-bias approaches leave little room (by definition) for alternative ways of incorpo- rating physics. The remaining two pathways encompass broader design spaces, which account for the observed heterogeneity in how the cor- responding methods are implemented. Although distinct schemes for introducing inductive bias have emerged, they share only broad concep- tual similarities. A similar pattern is observed 62 regarding learning bias, where prior knowledge is almost exclusively embedded via additional loss terms, though the form and weighting of these constraints vary widely. In parallel to these three pathways, hybrid approaches form a conceptually distinct fourth class, where a physics-based model and an ML model are either coupled in parallel or in series. Although PIML is generally subsumed underhybridin the context of PHM, distinguish- ing physics-informed from hybrid approaches elucidates the distinct ways in which prior knowledge is applied: the former integrate prior knowledge directly into the ML pipeline itself, whereas the latter retain an explicit stand-alone physics-based model that remains an integral part of the prediction pipeline during inference. Yet hybrid approaches constitute the least frequently adopted class. 2.Incorporation(b) How does the form of representation influence which approaches to incorporation are feasible? In terms of feasibility, the form in which prior physical knowledge is represented is not a minor implementation detail but the primary design lever that determines which pathways are viable. Mechanistic models are flexible enough to sup- port all four classes of approaches, while being most readily incorporated as observational bias or in a hybrid setting. Structural and topologi- cal information naturally maps to inductive bias via graph-based approaches, whereas qualitative properties are predominantly expressed as learn- ing bias and, in some cases, as simple architectural constraints. As indicated earlier, this creates a systematic trade-off: richer, more formal repre- sentations permit closer alignment with physics but require substantial modeling effort and are often asset-specific, whereas qualitative proper- ties are inherently limited in their physical fidelity yet typically retain broad applicability. Accord- ingly, with development largely being shaped by the form of representation, the effort to trans- form prior knowledge into suitable representations becomes instrumental in opening previously inac- cessible pathwaysâan aspect that has received little attention. 3.Practice(a) How does incorporating prior physical knowledge help overcome limitations of purely data-driven methods? Purely data-driven PHM applications con- tinue to face several challenges in real-world set- tings, including scarce and imbalanced degrada- tion data, sensitivity to distributional shift across operating conditions and assets, poor extrapo- lation beyond the training regime, and predic- tions that can be physically implausible. Although PIML is explicitly intended to mitigate these inherent shortcomings, empirical evidence only partially supports improvements in these areas. This does not imply that PIML is incapable of delivering improvements, but that the number of systematic experiments specifically targeting these areas remains limited across the literature. Nevertheless, across all four classes of approaches, the reviewed studies consistently demonstrate improved predictive performance for each PHM task and across a broad range of assets. There is also accumulating, though less robust, evidence that incorporating prior physical knowl- edge can enhance data efficiency, support more stable generalization, and enhance extrapola- tion. Furthermore, claims regarding improved interpretability, robustness, and physical consis- tency remain weakly substantiated: interpretabil- ity gains are typically inferred solely from the integration of prior knowledge but not quanti- fied, robustness is defined inconsistently and is rarely evaluated across a sufficiently broad range of operating and environmental conditions, and physical consistency lacks a shared definition and corresponding metrics, thereby leaving reported improvements largely qualitative. Lastly, some approaches are by definition structurally ill-suited to deliver on certain promises. Observational-bias methods, for exam- ple, can plausibly tackle data scarcity and improve in-distribution generalization, but they leave the hypothesis space unchanged and are therefore poorly equipped to enforce physical consistency or principled extrapolation beyond the regimes spanned by the (simulated) training data. Simi- larly, hybrid approaches that employ an in-series coupling, in which a physics-based model feeds an ML model, largely inherit the shortcomings attributed earlier to purely data-driven methods. 63 Overall, the current evidence indicates that physics-informed approaches already provide tan- gible advantages, yet broader benefits frequently ascribed to PIML in prior work are only par- tially substantiated and will require more rigorous, multi-dimensional evaluation before they can be regarded as confirmed. 3.Practice(b) What are the primary chal- lenges in developing and applying physics- informed approaches? Despite the promising outlook suggested by the reviewed literature, efforts to develop and apply PIML approaches within the field of PHM remain subject to major challenges. With the cur- rent landscape being highly fragmented, the field largely lacks standardized design patterns. Furthermore, each method is characterized by a certain degree of entanglement between the embedded physics and the specific asset addressedâan entanglement that determines its applicability to other settings. Given that the literature has largely focused on a limited set of assets (where the same problem-specific lim- itations apply), the field is still regarded as nascent in terms of empirically grounded devel- opment of transferable methods. The evidence suggests that continued incremental work within existing silos is unlikely to produce cumulative progress. Instead, advancement requires moving from problem-specific solutions toward founda- tional degradation models that facilitate physics- informed modeling irrespective of the asset in question. Furthermore,uncertaintyquantification, which is widely regarded as indispensable for risk- aware maintenance planning and safety-critical decision-making, is largely absent. Likewise, as noted earlier, the lack of evidence for inter- pretability impedes real-world adoption, since practitioners demand predictions that are not only accurate but also transparent and trustworthy. Lastly, online capability is rarely addressed explicitly, and closed-loop integration with exist- ing monitoring and maintenance workflows is vir- tually never demonstrated. Consequently, from a deployment perspective, most methods remain at the level of offline prototypes rather than oper- ational tools. Thus, the practical feasibility of large-scale industrial deployment has not yet been explored. Declarations FundingThe research leading to these results received funding from Deutsche Forschungsge- meinschaft (DFG, German Research Foundation) under grant number 514247199 as part of the research projectTheoMation. Conflict of InterestThe authors have no com- peting interests to declare that are relevant to the content of this article. References Abadi, H.(2023).Physics-informed deep learning-based approach for probabilistic modeling of degradation.Proceedings of the Annual Conference of the PHM Society 2023(Vol. 15). https://doi.org/10.36001/ phmconf.2023.v15i1.3806 Abiria, I., Wang, C., Zhang, Q., Liu, C., Jin, X. (2025). High-cycle and very-high-cycle fatigue life prediction in additive manufac- turing using hybrid physics-informed neural networks.Engineering Fracture Mechanics, 319. https://doi.org/10.1016/j.engfracmech .2025.111026 Aizpurua, J.I., Knutsen, K.E., Heimdal, M., Vanem, E. (2023). Integrated machine learn- ing and probabilistic degradation approach for vessel electric motor prognostics.Ocean Engineering,275. https://doi.org/10.1016/ j.oceaneng.2023.114153 Akkad, K. (2019). A physics based deep learn- ing technique for prognostics.Proceedings of the Annual Conference of the PHM Society 2019(Vol. 11). https://doi.org/10.36001/ phmconf.2019.v11i1.916 Arias Chao, M., Kulkarni, C., Goebel, K., Fink, O. (2022). Fusing physics-based and deep learning models for prognostics.Reliabil- ity Engineering and System Safety,217. https://doi.org/10.1016/j.ress.2021.107961 Atamuradov, V., Medjaher, K., Dersin, P., Lam- oureux, B., Zerhouni, N.(2017).Prog- nostics and health management for mainte- nance practitioners-review, implementation 64 and tools evaluation.International Jour- nal of Prognostics and Health Management, 8(3), 1â31. https://doi.org/10.36001/ijphm .2017.v8i3.2667 Bachar, L., & Bortman, J. (2024). A multi- disciplinary framework for vibration-based gear fault diagnosis using experiments, mod- eling, and machine learning.Proceedings of the Annual Conference of the PHM Society 2024(Vol. 16). https://doi.org/10.36001/ phmconf.2024.v16i1.4162 Badora, M., Bartosik, P., Graziano, A., Szolc, T.(2023).Using physics-informed neu- ral networks with small datasets to pre- dict the length of gas turbine nozzle cracks.Advanced Engineering Informat- ics,58. https://doi.org/10.1016/j.aei.2023 .102232 Bai, G., Su, Y., Rahman, M.M., Wang, Z. (2023). Prognostics of lithium-ion batteries using knowledge-constrained machine learn- ing and kalman filtering.Reliability Engi- neering and System Safety,231. https:// doi.org/10.1016/j.ress.2022.108944 Bajarunas, K., Baptista, M.L., Goebel, K., Chao, M.A.(2024).Health index estimation through integration of general knowledge with unsupervised learning.(Preprint at https://arxiv.org/abs/2405.04990v1) Baur, M., Albertelli, P., Monno, M. (2020). A review of prognostics and health manage- ment of machine tools.The International Journal of Advanced Manufacturing Tech- nology,107(5), 2843â2863. https://doi.org/ 10.1007/s00170-020-05202-3 Cai, S., Mao, Z., Wang, Z., Yin, M., Karni- adakis, G.E. (2021). Physics-informed neu- ral networks (PINNs) for fluid mechanics: A review.Acta Mechanica Sinica,37(12), 1727â1738. https://doi.org/10.1007/s10409 -021-01148-1 Cai, X., Zhang, D., Yu, Y., Xie, M. (2025). Knowl- edge embedded spatial-temporal graph con- volutional networks for remaining useful life prediction.Reliability Engineering and Sys- tem Safety,259. https://doi.org/10.1016/ j.ress.2025.110928 Cannizzaro, D., Antonioni, P., Ponzio, F., Galati, M., Patti, E., Cataldo, S.D.(2025). Machine learning-enabled real-time anomaly detection for electron beam powder bed fusion additive manufacturing.Journal of Intelligent Manufacturing,36(3), 2105â 2119.https://doi.org/10.1007/s10845-024 -02359-6 Carter, A., Imtiaz, S., Naterer, G. (2025). Imper- fect physics-guided neural networks.Chem- ical Engineering Science,305. https://doi .org/10.1016/j.ces.2024.121153 Champion, K., Lusch, B., Kutz, J.N., Brunton, S.L. (2019). Data-driven discovery of coor- dinates and governing equations.Proceed- ings of the National Academy of Sciences, 116(45), 22445â22451. https://doi.org/10 .1073/pnas.1906995116 Chao, Q., Hu, Y., Liu, C. (2025). Physics informed neural networks for detecting the wear of friction pairs in axial piston pumps.Relia- bility Engineering and System Safety,261. https://doi.org/10.1016/j.ress.2025.111144 Che, Y., Guo, J., Zheng, Y., Stroe, D.-I., Liu, W., Hu, X., Teodorescu, R. (2025). Unlocking interpretable prediction of battery random discharge capacity with domain adaptative physics constraint.Advanced Energy Mate- rials,15(13). https://doi.org/10.1002/aenm .202405506 Chen, C., Zhu, F., Xu, Z., Xie, Q., Lo, S.M., Tsui, K.L., Li, L.(2024).Knowledge- informed wheel wear prediction method for high-speed train using multisource signal data.IEEE Transactions on Instrumenta- tion and Measurement,73. https://doi.org/ 10.1109/TIM.2024.3413151 Chen, J., Tang, P., Rakstad, T., Patrick, M., Zhou, X. (2020). Augmenting a deep-learning algo- rithm with canal inspection knowledge for reliable water leak detection from multispec- tral satellite images.Advanced engineering informatics,46, 101161. https://doi.org/ 10.1016/j.aei.2020.101161 Chen, J., Wen, K., Xia, J., Huang, R., Chen, Z., Li, W.(2024).Knowledge embed- ded autoencoder network for harmonic drive fault diagnosis under few-shot industrial sce- narios.IEEE Internet of Things Journal, 11(13), 22915â22925.https://doi.org/10 .1109/JIOT.2024.3362343 Chen, L., Yang, F., Wang, R., Zhang, Y., Diao, Z., Rong, M. (2024). Optical spectral physics- informed attention network for condition monitoring in WAAM.IEEE Transactions 65 on Industrial Electronics,71(8), 9708â9718. https://doi.org/10.1109/TIE.2023.3325570 Chen, R.T.Q., Rubanova, Y., Bettencourt, J., Duvenaud, D.K. (2018). Neural ordinary differential equations. S. Bengio, H. Wal- lach, H. Larochelle, K. Grauman, N. Cesa- Bianchi, & R. Garnett (Eds.),Advances in Neural Information Processing Systems (Vol. 31). Curran Associates, Inc. Chen, Y., Rao, M., Feng, K., Zuo, M.J. (2022). Physics-informed LSTM hyperpa- rameters selection for gearbox fault detec- tion.Mechanical Systems and Signal Pro- cessing,171.https://doi.org/10.1016/j .ymssp.2022.108907 Chen, Z., Badrinarayanan, V., Lee, C.-Y., Rabi- novich, A. (2018). Gradnorm: Gradient normalization for adaptive loss balancing in deep multitask networks.International con- ference on machine learning(p. 794â803). Cheng, K., Zhang, K., Wang, Y., Yang, C., Li, J. (2024). Research on gas turbine health assessment method based on physical prior knowledge and spatial-temporal graph neu- ral network.Applied Energy,367. https:// doi.org/10.1016/j.apenergy.2024.123419 ClarivateAnalytics(2024).Journal citationreports.Retrievedfrom https://jcr.clarivate.com/jcr/home (Accessed: September 24, 2024) Cofre-Martel, S., Droguett, E.L., Modarres, M. (2020). A physics-informed deep learning approach for fatigue crack propagation.Pro- ceedings of the 30th European Safety and Reliability Conference and the 15th Proba- bilistic Safety Assessment and Management Conference(p. 2802).https://doi.org/10 .3850/978-981-14-8593-04973-cd Cuesta, J., Leturiondo, U., Vidal, Y., Pozo, F.(2025).A review of prognostics and health management techniques in wind energy.Reliability Engineering and System Safety,260. https://doi.org/10.1016/j.ress .2025.111004 Cuomo, S., Di Cola, V.S., Giampaolo, F., Rozza, G., Raissi, M., Piccialli, F. (2022). Scientific machine learning through physics-informed neural networks: Where we are and whatâs next.Journal of Scientific Computing, 92(3), 88. https://doi.org/10.1007/s10915 -022-01939-z Cvijic, S., Gupta, N., Lux, S. (2023). Need for AI in transformer diagnostics and prognostics. 2023 Annual Reliability and Maintainabil- ity Symposium (RAMS).https://doi.org/ 10.1109/RAMS51473.2023.10088199 Deng, W., Nguyen, K., Medjaher, K., Gogu, C., Morio, J. (2023). Physics-informed machine learning in prognostics and health man- agement: State of the art and challenges. Applied Mathematical Modelling,124, 325â 352. https://doi.org/10.1016/j.apm.2023.07 .011 Deng, Y., Du, C., Ren, Z.(2025).A novel method for estimating the state of health of lithium-ion batteries based on physics- informed neural network.Batteries,11(2). https://doi.org/10.3390/batteries11020049 Deng, Y., Du, S., Wang, D., Shao, Y., Huang, D. (2023). A calibration-based hybrid trans- fer learning framework for RUL prediction of rolling bearing across different machines. IEEE Transactions on Instrumentation and Measurement,72. https://doi.org/10.1109/ TIM.2023.3260283 Dong, C., Tao, J., Sun, H., Wei, Q., Tan, H., Liu, C. (2025). Innovative fault diagnosis for axial piston pumps: A physics-informed neural network framework predicting pump flow ripple.Mechanical Systems and Signal Processing,225. https://doi.org/10.1016/ j.ymssp.2024.112274 Dong, Y., Li, Y., Zheng, H., Wang, R., Xu, M. (2022). A new dynamic model and trans- fer learning based intelligent fault diagnosis framework for rolling element bearings race faults: Solving the small sample problem. ISA Transactions,121, 327â348. https:// doi.org/10.1016/j.isatra.2021.03.042 Dourado, A., & Viana, F.(2019).Physics- informed neural networks for corrosion- fatigue prognosis.Proceedings of the Annual Conference of the PHM Society 2019(Vol. 11). https://doi.org/10.36001/ phmconf.2019.v11i1.814 Dourado, A., & Viana, F. (2022). Ensemble of hybrid neural networks to compensate for epistemic uncertainties: a case study in sys- tem prognosis.Soft Computing,26(13), 6157â6173. https://doi.org/10.1007/s00500 -022-07129-1 66 Dwivedi, D., Yemula, P.K., Pal, M.(2023). DynamoPMU: A physics informed anomaly detection and prediction methodology using non-linear dynamics fromÎźPMU measure- ment data.(Preprint at https://arxiv.org/ abs/2304.00092v1) E, L., Wang, J., Yang, R., Wang, C., Li, H., Xiong, R.(2025).A physics-informed neural network-based method for predicting degradation trajectories and remaining use- ful life of supercapacitors.Green Energy and Intelligent Transportation,4(3). https:// doi.org/10.1016/j.geits.2025.100291 Ellis, B., Heyns, P.S., Schmidt, S. (2022). A hybrid framework for remaining useful life estimation of turbomachine rotor blades. Mechanical Systems and Signal Processing, 170. https://doi.org/10.1016/j.ymssp.2022 .108805 Exenberger, J., Di Salvo, M., Hirsch, T., Wotawa, F., Schweiger, G.(2024). Generalizabletemperaturenowcast- ingwithphysics-constrainedRNNs forpredictivemaintenanceofwind turbinecomponents.(Preprintat https://arxiv.org/abs/2404.04126v1) Farhat, H., & Altarawneh, A. (2025). Physics- informed machine learning for intelligent gas turbine digital twins: A review.Ener- gies,18(20), 5523. https://doi.org/10.3390/ en18205523 Fassi, Y., Heiries, V., Boutet, J., Boisseau, S.(2024a).Physics-informed machine learning for robust remaining useful life estimation of power MOSFETs.2024 IEEE International Conference on Prog- nostics and Health Management (ICPHM) (p. 399â406).https://doi.org/10.1109/ ICPHM61352.2024.10626501 Fassi, Y., Heiries, V., Boutet, J., Boisseau, S. (2024b). Toward physics-informed machine- learning-based predictive maintenance for power converters-a review.IEEE Transac- tions on Power Electronics,39(2), 2692â 2720. https://doi.org/10.1109/TPEL.2023 .3328438 Feng, X., Zhang, Y., Xiong, R., Wang, C. (2024).Comprehensive performance com- parison among different types of features in data-driven battery state of health estima- tion.(Preprint at https://arxiv.org/abs/ 2308.13993v1) Feng, Y., Chen, J., Liu, Z., Lv, H., Wang, J.(2022).Full graph autoencoder for one-class group anomaly detection of IIoT system.IEEE Internet of Things Jour- nal,9(21), 21886â21898. https://doi.org/ 10.1109/JIOT.2022.3181737 Fern Ěandez, J., Chiach ĚÄąo, J., Barros, J., Chiach ĚÄąo, M., Kulkarni, C.S.(2024).Physics- guided recurrent neural network trained with approximate bayesian computation: A case study on structural response prognos- tics.Reliability Engineering and System Safety,243. https://doi.org/10.1016/j.ress .2023.109822 Fern Ěandez, J., Corbetta, M., Kulkarni, C.S., Chi- ach ĚÄąo, J., Chiach ĚÄąo, M. (2024). Training of physics-informed bayesian neural networks with ABC-S for prognostic of li-ion batter- ies.Computers in Industry,155. https:// doi.org/10.1016/j.compind.2023.104058 Firoozi, R., Sattarzadeh, S., Dey, S. (2022). Cylin- drical battery fault detection under extreme fast charging: A physics-based learning approach.IEEE Transactions on Energy Conversion,37(2), 1241â1250. https://doi .org/10.1109/TEC.2021.3112950 Freeman, B., Tang, Y., Huang, Y., Vanzwieten, J.(2022).Physics-informed turbulence intensity infusion: A new hybrid approach for marine current turbine rotor blade fault detection.Ocean Engineering,254. https:// doi.org/10.1016/j.oceaneng.2022.111299 Fricke, K.(2023).Mission-specific progno- sis of li-ion batteries using hybrid physics- informed neural networks.Proceedings of the Annual Conference of the PHM Society 2023(Vol. 15). https://doi.org/10.36001/ phmconf.2023.v15i1.3796 Fu, H., Liu, Z., Cui, K., Du, Q., Wang, J., Shi, D. (2024). Physics-informed neural network for spacecraft lithium-ion battery modeling and health diagnosis.IEEE/ASME Trans- actions on Mechatronics, 1â10.https:// doi.org/10.1109/TMECH.2023.3348519 Furlong, T., & Reichard, K.(2023).A physics-informed, transfer learning approach to structural health monitoring.Proceed- ings of the Annual Conference of the PHM 67 Society 2023(Vol. 15). https://doi.org/10 .36001/phmconf.2023.v15i1.3802 Gao, C., Wang, Z., Guo, Y., Wang, H., Yi, H. (2024). MPINet: Multiscale physics- informed network for bearing fault diag- nosis with small samples.IEEE Trans- actions on Industrial Informatics,20(12), 14371â14380. https://doi.org/10.1109/TII .2024.3452174 Gao, F., Zhang, S., You, Z., Cao, S. (2024). Fault diagnosis of AC transmission system via GoogleNet.2024 IEEE 2nd International Conference on Power Science and Technol- ogy (ICPST)(p. 1052-1057). https://doi .org/10.1109/ICPST61417.2024.10602407 Gao, J., Zheng, D., Yang, S.(2021).Sens- ing the disturbed rhythm of city mobility with chaotic measures: anomaly awareness from traffic flows.Journal of Ambient Intel- ligence and Humanized Computing,12(4), 4347â4362. https://doi.org/10.1007/s12652 -019-01338-7 Gareev, A., Protsenko, V., Stadnik, D., Gresh- niakov, P., Yuzifovich, Y., Minaev, E., . . . Nikonorov, A. (2021). Improved fault diag- nosis in hydraulic systems with gated convo- lutional autoencoder and partially simulated data.Sensors,21(13).https://doi.org/ 10.3390/s21134410 Garpelli, L.N., Alves, D.S., Cavalca, K.L., de Castro, H.F. (2023). Physics-guided neu- ral networks applied in rotor unbalance problems.Structural Health Monitoring, 22(6), 4117â4130. https://doi.org/10.1177/ 14759217231163081 Ge, L., & Sadhu, A. (2024). Domain adaptation for structural health monitoring via physics- informed and self-attention-enhanced gener- ative adversarial learning.Mechanical Sys- tems and Signal Processing,211. https:// doi.org/10.1016/j.ymssp.2024.111236 Gij Ěon, A., Pujana-Goitia, A., Perea, E., Molina- Solana, M., G Ěomez-Romero, J. (2023).Pre- diction of wind turbines power with physics- informed neural networks and evidential uncertainty quantification.(Preprint at https://arxiv.org/abs/2307.14675v1) Gong, H., Baffour, F.I., Glazebrook, K.N., Rhodes, N.G., Tiegs-Heiden, C.A., Thorne, J.E., . . . others(2022).Deep learning- based virtual noncalcium imaging in multi- ple myeloma using dual-energy CT.Med- ical physics,49(10), 6346â6358. https:// doi.org/10.1002/mp.15934 Goodman, D., Hofmeister, J.P., Szidarovszky, F. (2019).Prognostics and health management: A practical approach to improving system reliability using condition-based data. John Wiley & Sons.https://doi.org/10.1002/ 9781119356677 Gouriveau, R., Medjaher, K., Zerhouni, N. (2016). From prognostics and health systems man- agement to predictive maintenance 1: Mon- itoring and prognostics. John Wiley & Sons. https://doi.org/10.1002/9781119371052 Guo, D., Yang, G., Han, X., Feng, X., Lu, L., Ouyang, M. (2021). Parameter identifica- tion of fractional-order model with trans- fer learning for aging lithium-ion batteries. International Journal of Energy Research, 45(9), 12825â12837.https://doi.org/10 .1002/er.6614 Gurgen, A., & Dinh, N.T. (2022). Development and assessment of a reactor system prog- nosis model with physics-guided machine learning.Nuclear Engineering and Design, 398.https://doi.org/10.1016/j.nucengdes .2022.111976 Hagmeyer, S., Zeiler, P., Huber, M.F. (2022). On the integration of fundamental knowl- edge about degradation processes into data- driven diagnostics and prognostics using theory-guided data science.PHM Soci- ety European Conference(Vol. 7, p. 156â 165). https://doi.org/10.36001/phme.2022 .v7i1.3352 Hajiha, M., Liu, X., Lee, Y.M., Ramin, M. (2022). A physics-regularized data-driven approach for health prognostics of complex engineered systems with dependent health states.Reli- ability Engineering and System Safety,226. https://doi.org/10.1016/j.ress.2022.108677 Han, G., Chen, J., Liu, L., Wang, Z., Zhang, F., Abudurexiti, Y. (2024). An interpretable CNN with wavelet group policy embedded for intelligent fault diagnosis.IEEE Trans- actions on Instrumentation and Measure- ment,73, 1â15. https://doi.org/10.1109/ TIM.2024.3368479 68 Han, S., Awasthi, U., Bollas, G.M.(2025). Physics-informed symbolic regression for tool wear and remaining useful life predic- tions in manufacturing.Journal of Man- ufacturing Systems,80, 734â748. https:// doi.org/10.1016/j.jmsy.2025.03.023 Hao, C., Mao, X., Ma, T., He, S., Li, B., Liu, H., . . . Zhang, L. (2023). A novel deep learning method with partly explainable: Intelligent milling tool wear prediction model based on transformer informed physics.Advanced Engineering Informatics,57. https://doi .org/10.1016/j.aei.2023.102106 He, G., Zhao, Y., Yan, C.(2023).Multi- axial fatigue life prediction using physics- informed neural networks with sensitive fea- tures.Engineering Fracture Mechanics, 289. https://doi.org/10.1016/j.engfracmech .2023.109456 He, X., Li, L., Wang, Y., Zheng, H., Cao, K., Yan, K., . . . Zhou, M. (2024).Training-free large model priors for multiple-in-one image restoration.(Preprint at https://arxiv.org/ abs/2407.13181v1) He, Y., Su, H., Zio, E., Peng, S., Fan, L., Yang, Z., Zhang, J. (2023). A systematic method of remaining useful life estimation based on physics-informed graph neural networks with multisensor data.Reliability Engineer- ing and System Safety,237. https://doi .org/10.1016/j.ress.2023.109333 He, Z., Wang, S., Shi, J., Liu, D., Duan, X., Shang, Y. (2025). Physics-informed neural network supported Wiener process for degradation modeling and reliability prediction.Relia- bility Engineering and System Safety,258. https://doi.org/10.1016/j.ress.2025.110906 Herv Ěe de Beaulieu, M., Jha, M.S., Garnier, H., Cerbah, F.(2024).Remaining useful life prediction based on physics-informed data augmentation.Reliability Engineering and System Safety,252. https://doi.org/ 10.1016/j.ress.2024.110451 Hong, R., Nie, S., Ji, H., Ma, Z.(2025). Physics-informed machine learning for high- speed on/off valve performance degrada- tion prediction in water hydraulic manipu- lator.Reliability Engineering and System Safety,261. https://doi.org/10.1016/j.ress .2025.111106 Hu, Y., Miao, X., Si, Y., Pan, E., Zio, E. (2022).Prognostics and health manage- ment: A review from the perspectives of design, development and decision.Reliability Engineering & System Safety,217, 108063. https://doi.org/10.1016/j.ress.2021.108063 Hu, Y., Zhang, X., Zou, X., Sun, M., Zheng, Y., Min, G.(2017).Semi-supervised speech enhancement combining nonnega- tive matrix factorization and robust prin- cipal component analysis.IEICE Trans- actions on Fundamentals of Electronics, Communications and Computer Sciences, 100(8), 1714â1719.https://doi.org/10 .1587/transfun.E100.A.1714 Huang, C., Bu, S., Lee, H.H., Chan, C.H., Kong, S.W., Yung, W.K. (2024). Prognostics and health management for predictive mainte- nance: A review.Journal of Manufacturing Systems,75, 78â101.https://doi.org/10 .1016/j.jmsy.2024.05.021 Huang, H., Meng, J., Wang, Y., Cai, L., Peng, J., Wu, J., . . . Teodorescu, R. (2022). An enhanced data-driven model for lithium-ion battery state-of-health estimation with opti- mized features and prior knowledge.Auto- motive Innovation,5(2), 134â145. https:// doi.org/10.1007/s42154-022-00175-3 Huang, Y., Tang, B., Yang, Q., Ming, Z. (2025). Physics-informed causal learning network for fault diagnosis of rotating machinery under unseen operating conditions.Neuro- computing,639. https://doi.org/10.1016/ j.neucom.2025.130187 Jahani-Nasab, M., & Bijarchi, M.A.(2024). Enhancing convergence speed with fea- ture enforcing physics-informed neural net- works using boundary conditions as prior knowledge.Scientific Reports,14(1), 23836. https://doi.org/10.1038/s41598-024 -74711-y Jang, J., Jo, J., Kim, J., Lee, S., Lee, T., Yoo, J. (2025). State of health estimation of lithium- ion battery cell based on optical thermom- etry with physics-informed machine learn- ing.Engineering Applications of Artificial Intelligence,140. https://doi.org/10.1016/ j.engappai.2024.109704 Jia, X., Huang, B., Feng, J., Cai, H., Lee, J. (2018).A review of PHM data compe- titions from 2008 to 2017: Methodologies and analytics.Proceedings of the Annual 69 Conference of the Prognostics and Health Management Society(p. 1â10). https:// doi.org/10.36001/phmconf.2018.v10i1.462 Jia, X., Zhang, C., Li, Y., Zou, C., Wang, L.Y., Cai, X. (2024). Knee-point-conscious bat- tery aging trajectory prediction based on physics-guided machine learning.IEEE Transactions on Transportation Electrifica- tion,10(1), 1056â1069. https://doi.org/10 .1109/TTE.2023.3266386 Jiang, C., Zhong, T., Choi, H., Youn, B.D. (2025). Physics-informed Gaussian process proba- bilistic modeling with multi-source data for prognostics of degradation processes.Relia- bility Engineering and System Safety,258. https://doi.org/10.1016/j.ress.2025.110893 Jiang, F., Hou, X., Xia, M.(2024).Spatio- temporal attention-based hidden physics- informed neural network for remaining use- ful life prediction.(Preprint at https:// arxiv.org/abs/2405.12377v1) Jin, X., Lv, S., Kong, Z., Yang, H., Zhang, Y., Guo, Y., Xu, Z. (2024). Graph spatio- temporal networks for condition monitoring of wind turbine.IEEE Transactions on Sustainable Energy, 1â11. https://doi.org/ 10.1109/TSTE.2024.3411884 Kadiwala, S., Savsaviya, P., Pandey, S.V., Singh, A.K., Prochowicz, D., Akin, S., . . . Yadav, P. (2025). Decoding degradation: The syn- ergy of partial differential equations and advanced predictive models for lithium- ion battery.Journal of Power Sources, 627.https://doi.org/10.1016/j.jpowsour .2024.235771 Karniadakis, G.E., Kevrekidis, I.G., Lu, L., Perdikaris, P., Wang, S., Yang, L. (2021). Physics-informed machine learning.Nature Reviews Physics,3(6), 422â440. https:// doi.org/10.1038/s42254-021-00314-5 Karpatne, A., Atluri, G., Faghmous, J.H., Stein- bach, M., Banerjee, A., Ganguly, A., . . . Kumar, V. (2017). Theory-guided data sci- ence: A new paradigm for scientific discovery from data.IEEE Transactions on knowl- edge and data engineering,29(10), 2318â 2331. https://doi.org/10.1109/TKDE.2017 .2720168 Kayedpour, N., Qing, J., Wauters, J., de Koon- ing, J., Couckuyt, I., Crevecoeur, G. (2024). Wind turbine hybrid physics-based deep learning model for a health monitoring approach considering provision of ancillary services.IEEE Transactions on Instrumen- tation and Measurement,73, 1â14. https:// doi.org/10.1109/TIM.2024.3375416 Keizers, L.S., Loendersloot, R., Tinga, T. (2021). Unscented kalman filtering for prognos- tics under varying operational and envi- ronmental conditions.International Jour- nal of Prognostics and Health Management, 12(2). https://doi.org/10.36001/ijphm.2021 .v12i2.2943 Khan, S., Yairi, T., Tsutsumi, S., Nakasuka, S. (2024). A review of physics-based learn- ing for system health management.Annual Reviews in Control,57. https://doi.org/ 10.1016/j.arcontrol.2024.100932 Kim, I., Wook Kim, S., Kim, J., Huh, H., Jeong, I., Choi, T., Lee, S. (2024). Single domain generalizable and physically interpretable bearing fault diagnosis for unseen working conditions.Expert Systems with Applica- tions,241. https://doi.org/10.1016/j.eswa .2023.122455 Kim, N.-H., An, D., Choi, J.-H. (2017). Prognos- tics and health management of engineering systems.Switzerland: Springer Interna- tional Publishing. https://doi.org/10.1007/ 978-3-319-44742-1 Kobrich, P., Martin, G.S., Droguett, E.L., Bernardin, A.O., Ayele, Y.Z.(2020). Physics based deep learning model for crack propagation prognostics.Proceedings of the 29th European Safety and Reliability Con- ference(p. 1236â1241). https://doi.org/ 10.3850/978-981-11-2724-30323-cd Kohtz, S., & Wang, P. (2022). Physics-based machine learning with filtering for fail- ure prognostics partially observable dynamic systems.2022 Annual Reliability and Maintainability Symposium (RAMS)(Vol. 2022-January).https://doi.org/10.1109/ RAMS51457.2022.9893922 Kohtz, S., Xu, Y., Zheng, Z., Wang, P. (2022). Physics-informed machine learning model for battery state of health prognostics using partial charging segments.Mechanical Sys- tems and Signal Processing,172. https:// doi.org/10.1016/j.ymssp.2022.109002 Koutsoupakis, J., Seventekidis, P., Giagopoulos, 70 D. (2023). Machine learning based condition monitoring for gear transmission systems using data generated by optimal multibody dynamics models.Mechanical Systems and Signal Processing,190. https://doi.org/10 .1016/j.ymssp.2023.110130 Krizhevsky, A., Sutskever, I., Hinton, G.E. (2012). Imagenet classification with deep convolu- tional neural networks.Advances in neural information processing systems,25 Kulkarni, C.S., Biswas, G., Celaya, J.R., Goebel, K.(2013).Physics based degradation models for electrolytic capacitor prognostics under thermal overstress conditions.Inter- national Journal of Prognostics and Health Management,4(1) Kumar,G.,Vetrivelan,P.,Kumba,K., Ajeyprasaath, K.B.(2023).Estimat- ing remaining useful life and state of health of EV lithium ion batteries using sequential CNNs.2023 Innovations in Power and Advanced Computing Technolo- gies (i-PACT).https://doi.org/10.1109/ I-PACT58649.2023.10434881 Kumari, L.N., & Wang, P.(2024).Effi- cient stochastic parametric estimation for lithium-ion battery performance degrada- tion tracking and prognosis.Journal of Manufacturing Systems,75, 270â277. https://doi.org/10.1016/j.jmsy.2024.03.017 Kundu, P., Darpe, A.K., Kulkarni, M.S. (2020). A review on diagnostic and prognostic approaches for gears.Structural Health Monitoring,20(5), 2853â2893. https://doi .org/10.1177/1475921720972926 Kundu, P., Darpe, A.K., Kulkarni, M.S. (2024). Development of data-driven, physics-based, and hybrid prognosis frameworks: a case study for gear remaining useful life pre- diction.Journal of Intelligent Manufac- turing. https://doi.org/10.1007/s10845-024 -02477-1 Lai, C., Baraldi, P., Zio, E. (2024). Physics- informed deep autoencoder for fault detec- tion in new-design systems.Mechanical Sys- tems and Signal Processing,215. https:// doi.org/10.1016/j.ymssp.2024.111420 Lee, H., Kang, E., Kim, D., Yoon, J. (2023). Development of the estimation model for the maximum power point of building-applied photovoltaic systems based on machine learning.Journal of Building Engineer- ing,76.https://doi.org/10.1016/j.jobe .2023.107285 Lehmann, T., Berendes, E., Kratzing, R., Sethia, G. (2024). Learning the ageing behaviour of lithium-ion batteries using electric vehicle fleet analysis.Batteries,10(12). https:// doi.org/10.3390/batteries10120432 Lei, Z., Zhang, P., Chen, Y., Feng, K., Wen, G., Liu, Z., . . . Yang, C. (2023). Prior knowledge-embedded meta-transfer learning for few-shot fault diagnosis under variable operating conditions.Mechanical Systems and Signal Processing,200.https://doi .org/10.1016/j.ymssp.2023.110491 Leng, J., Zuo, K., Xu, C., Zhou, X., Zheng, S., Kang, J., . . . Gao, R.X.(2026). Physics-informed machine learning in intel- ligent manufacturing: a review.Journal of Intelligent Manufacturing,37(6), 2215â 2257.https://doi.org/10.1007/s10845-025 -02641-1 Li, B., Zhu, J., Zhao, X.(2025).A hybrid physics informed predictive scheme for pre- dicting low-cycle fatigue life and reliabil- ity of aerospace materials under multiaxial loading conditions.Reliability Engineering and System Safety,257. https://doi.org/ 10.1016/j.ress.2025.110838 Li, C., Zhai, W., Fu, W., Qin, J., Kang, Y. (2025). Remaining useful life prediction of rolling bearings based on parallel feature extrac- tion.Robotic Intelligence and Automation, 45(1), 90â105.https://doi.org/10.1108/ RIA-03-2024-0061 Li, H., Zhang, Z., Li, T., Si, X. (2024). A review on physics-informed data-driven remaining useful life prediction: Challenges and oppor- tunities.Mechanical Systems and Signal Processing,209. https://doi.org/10.1016/ j.ymssp.2024.111120 Li, R., Xia, T., Luo, F., Jiang, Y., Chen, Z., Xi, L. (2024). Hybrid physics-embedded recurrent neural networks for fault diagnosis under time-varying conditions based on mul- tivariate proprioceptive signals.Advanced Engineering Informatics,62. https://doi .org/10.1016/j.aei.2024.102851 Li, S., Li, J., Zhu, K.(2025).Application of physics-guided deep learning model in tool wear monitoring of high-speed milling. 71 Mechanical Systems and Signal Processing, 224. https://doi.org/10.1016/j.ymssp.2024 .111949 Li, S., Lin, X., Shi, H., Shi, Y., Zhu, K. (2024). Physics-guided deep learning method for tool condition monitoring in smart machin- ing system.IEEE/ASME Transactions on Mechatronics,29(3), 2327â2337. https:// doi.org/10.1109/TMECH.2023.3311435 Li, T. (2024). Particle filter-based fatigue dam- age prognosis using prognostic-aided model updating.Mechanical Systems and Signal Processing,211. https://doi.org/10.1016/ j.ymssp.2024.111244 Li, T., Chen, J., Yuan, S., Zarouchas, D., Sbarufatti, C., Cadini, F. (2024). Parti- cle filter-based fatigue damage prognosis byfusingmultipledegradationmod- els.Structural Health Monitoring,23(5), 3253â3275.https://doi.org/10.1177/ 14759217231216697 Li, T., Sbarufatti, C., Cadini, F., Chen, J., Yuan, S. (2021). Particle filter-based hybrid dam- age prognosis considering measurement bias. Structural Control and Health Monitoring, 29(4). https://doi.org/10.1002/stc.2914 Li, Y., Wang, T., Xie, J., Yang, J., Pan, T., Yang, B. (2024). A simulation data-driven semi-supervised framework based on MK- KNN graph and ESSGAT for bearing fault diagnosis.ISA Transactions,155, 261â 273.https://doi.org/10.1016/j.isatra.2024 .09.029 Liang, J., Liu, H., Xiao, N.-C. (2024). A hybrid approach based on deep neural network and double exponential model for remaining use- ful life prediction.Expert Systems with Applications,249. https://doi.org/10.1016/ j.eswa.2024.123563 Liao, J.-X., He, C., Li, J., Sun, J., Zhang, S., Zhang, X. (2025). Classifier-guided neu- ral blind deconvolution: A physics-informed denoising module for bearing fault diagnosis under noisy conditions.Mechanical Sys- tems and Signal Processing,222. https:// doi.org/10.1016/j.ymssp.2024.111750 Liao, X., Chen, S., Wen, P., Zhao, S. (2023). Remaining useful life with self-attention assisted physics-informed neural network. Advanced Engineering Informatics,58. https://doi.org/10.1016/j.aei.2023.102195 Lin, C., Tuo, X., Wu, L., Zhang, G., Lyu, Z., Zeng, X. (2025). Physics-informed machine learn- ing for accurate SOH estimation of lithium- ion batteries considering various tempera- tures and operating conditions.Energy, 318. https://doi.org/10.1016/j.energy.2025 .134937 Liu, S., Chen, J., Liu, Z., Wang, J., Wang, Z.J. (2025). Graph embedded patch-sense autoencoder with prior knowledge for multi- component system anomaly detection.Reli- ability Engineering and System Safety,256. https://doi.org/10.1016/j.ress.2024.110784 Liu, X., Cai, H., Zhou, Z., Kong, Y., Zhou, X., Han, X., . . . Zheng, Y. (2024). Enhanc- ing multi-type fault diagnosis in lithium-ion battery systems: Vision transformer-based transfer learning approach.Journal of Power Sources,624.https://doi.org/10 .1016/j.jpowsour.2024.235610 Liu, Y., Chen, H., Yao, L., Ding, J., Chen, S., Wang, Z. (2025). A physics-guided approach for accurate battery SOH estimation using RCMHCRE and BatteryPINN.Advanced Engineering Informatics,65. https://doi .org/10.1016/j.aei.2025.103211 Liu, Y., Wang, T., Chu, F. (2023). Knowledge embedded lightweight vision transformer for machine condition monitoring.Measure- ment: Journal of the International Measure- ment Confederation,221. https://doi.org/ 10.1016/j.measurement.2023.113402 Liu, Y., Zhou, B., Pang, T., Fan, G., Zhang, X.(2024).Hybrid fusion for battery degradation diagnostics using minimal real- world data: Bridging laboratory and practi- cal applications.(Preprint at https://arxiv .org/abs/2410.06469v1) Liu, Z., Liu, Y., Zhang, D., Cai, B., Zheng, C. (2015). Fault diagnosis for a solar assisted heat pump system under incomplete data and expert knowledge.Energy,87, 41â 48.https://doi.org/10.1016/j.energy.2015 .04.090 Lu, F., Tong, Q., Jiang, X., Du, X., Xu, J., Huo, J.(2025).Prior knowledge embedding convolutional autoencoder: A single-source domain generalized fault diagnosis frame- work under small samples.Computers in 72 Industry,164.https://doi.org/10.1016/j .compind.2024.104169 Lu, L., Jin, P., Pang, G., Zhang, Z., Karni- adakis, G.E.(2021).Learning nonlin- ear operators via DeepONet based on the universal approximation theorem of oper- ators.Nature machine intelligence,3(3), 218â229.https://doi.org/10.1038/s42256 -021-00302-5 Luo, Y., Ju, S., Li, P., Zhang, H. (2025). A method for estimating lithium-ion battery state of health based on physics-informed hybrid neural network.Electrochimica Acta, 525.https://doi.org/10.1016/j.electacta .2025.146110 Lyathakula, K.R., & Yuan, F.-G.(2021). A probabilistic fatigue life prediction for adhesively bonded joints via ANNs-based hybrid model.International Journal of Fatigue,151.https://doi.org/10.1016/j .ijfatigue.2021.106352 Ma, L., Tian, J., Zhang, T., Guo, Q., Hu, C. (2024).Accurate and efficient remaining useful life prediction of batteries enabled by physics-informed machine learning.Journal of Energy Chemistry,91, 512â521. https:// doi.org/10.1016/j.jechem.2023.12.043 Ma, Z., Fu, L., Xu, F., Zhang, L.(2025). A physics-based sample generation method for few-shot bearing condition monitoring. Knowledge-Based Systems,310. https:// doi.org/10.1016/j.knosys.2024.112952 Ma, Z., Liao, H., Gao, J., Nie, S., Geng, Y.(2023).Physics-informed machine learning for degradation modeling of an electro-hydrostatic actuator system.Relia- bility Engineering and System Safety,229. https://doi.org/10.1016/j.ress.2022.108898 Ma, Z., Zhao, M., Dai, X., Chen, Y.(2023). Ahybrid-drivenprobabilisticstate space model for tool wear monitoring. MechanicalSystemsandSignalPro- cessing,200.https://doi.org/10.1016/ j.ymssp.2023.110599 Mammeri, S., Barros, B., Conde-Carnero, B., Riveiro, B. (2025). From traditional dam- age detection methods to physics-informed machine learning in bridges: A review.Engi- neering Structures,330, 119862. https:// doi.org/10.1016/j.engstruct.2025.119862 Matania, O., Bachar, L., Khemani, V., Das, D., Azarian, M.H., Bortman, J. (2023). One- fault-shot learning for fault severity esti- mation of gears that addresses differences between simulation and experimental sig- nals and transfer function effects.Advanced Engineering Informatics,56. https://doi .org/10.1016/j.aei.2023.101945 Mauthe, F., Braun, C., Raible, J., Zeiler, P., Huber, M.F.(2024).Overview of pub- licly available degradation data sets for tasks within prognostics and health manage- ment.(Preprint at https://arxiv.org/abs/ 2403.13694) Mauthe, F., Steinmann, L., Neu, M., Zeiler, P.(2025).Overview and analysis of publicly available degradation data sets for tasks within prognostics and health management.35th european safety and reliability conference. Research Publish- ing.https://doi.org/10.3850/978-981-94 -3281-3ESREL-SRA-E2025-P7412-cd McMahon, C.A., Roberts, D.A., Stella, J.C., Trugman, A.T., Singer, M.B., Caylor, K.K. (2024). A river runs through it: Robust automated mapping of riparian woodlands and land surface phenology across dryland regions.Remote sensing of environment, 305, 114056. https://doi.org/10.1016/j.rse .2024.114056 Mei, F., Chen, H., Yang, W., Zhai, G. (2024). A hybrid physics-informed machine learn- ing approach for time-dependent reliability assessment of electromagnetic relays.Relia- bility Engineering and System Safety,252. https://doi.org/10.1016/j.ress.2024.110385 Meng, H., & Li, Y.-F.(2019).A review on prognostics and health management (PHM) methods of lithium-ion batteries.Renew- able and Sustainable Energy Reviews,116. https://doi.org/10.1016/j.rser.2019.109405 Mienye, I.D., & Swart, T.G. (2024). A comprehen- sive review of deep learning: Architectures, recent advances, and applications.Informa- tion,15(12), 755. https://doi.org/10.3390/ info15120755 Mochammad, S., Noh, Y., Kim, N.H.(2023). Enhancing realistic remaining useful life prediction through multi-fidelity physics- informed approaches.Proceedings of the Annual Conference of the PHM Society 2023 73 (Vol. 15). Najera-Flores, D.A., Hu, Z., Chadha, M., Todd, M.D.(2023).A physics-constrained bayesian neural network for battery remain- ing useful life prediction.Applied Mathemat- ical Modelling,122, 42â59. https://doi.org/ 10.1016/j.apm.2023.05.038 Nascimento, R.G., Corbetta, M., Kulkarni, C.S., Viana, F. (2021). Hybrid physics-informed neural networks for lithium-ion battery modeling and prognosis.Journal of Power Sources,513.https://doi.org/10.1016/j .jpowsour.2021.230526 Navidi, S., Thelen, A., Li, T., Hu, C. (2023). Physics-informed neural networks for degra- dation diagnostics of lithium-ion batteries. ASME 2023 International Design Engineer- ing Technical Conferences and Computers and Information in Engineering Confer- ence(Vol. 3A).https://doi.org/10.1115/ DETC2023-116940 Navidi, S., Thelen, A., Li, T., Hu, C. (2024). Physics-informed machine learning for bat- tery degradation diagnostics: A comparison of state-of-the-art methods.(Preprint at https://arxiv.org/abs/2404.04429v1) Nectoux, P., Gouriveau, R., Medjaher, K., Ramasso, E., Morello, B., Zerhouni, N., Varnier, C.(2012).PRONOSTIA: An experimental platform for bearings acceler- ated life test.IEEE International Confer- ence on Prognostics and Health Manage- ment.Denver, CO, USA. Neerukatti, R.K., Chattopadhyay, A., Iyyer, N., Phan, N.(2018).A hybrid prognosis model for predicting fatigue crack propa- gation under biaxial in-phase and out-of- phase loading.Structural Health Moni- toring,17(4), 888â901. https://doi.org/10 .1177/1475921717725019 Nguyen, R., Singh, S.K., Rai, R. (2023). Physics- infused fuzzy generative adversarial network for robust failure prognosis.Mechanical Sys- tems and Signal Processing,184. https:// doi.org/10.1016/j.ymssp.2022.109611 Pan, S., Li, P., Zeng, D., Guo, S., Hu, G. (2019). A Q-Learning based framework for con- gested link identification [Article].IEEE Internet of Things Journal,6(6), 9668 â 9678.https://doi.org/10.1109/JIOT.2019 .2930459 Pan, Y., Sharif Khodaei, Z., Aliabadi, F. (2025).In-service fatigue crack monitor- ing through baseline-free automated detec- tion and physics-informed neural network quantification.NDT and E International, 153. https://doi.org/10.1016/j.ndteint.2025 .103360 Pardoe, D., & Stone, P.(2010).Boosting for regression transfer.Proceedings of the 27th International Conference on Interna- tional Conference on Machine Learning(p. 863â870). Pettorossi, C., Heiries, V., Gerard, M., Rosini, S.(2024).Addressing data scarcity in PEMFC fault diagnosis using adversarial learning.2024 IEEE International Con- ference on Prognostics and Health Manage- ment (ICPHM)(p. 393â398). https://doi .org/10.1109/ICPHM61352.2024.10627464 Pettorossi, C., Morvillier, R., Heiries, V., Rosini, S., Gerard, M.(2025).Physics-guided fault diagnosis method for proton exchange membrane fuel cells based on LSTM neu- ral network.Journal of Power Sources, 626.https://doi.org/10.1016/j.jpowsour .2024.235696 Pugalenthi, K., Park, H., Hussain, S., Raghavan, N. (2024). Remaining useful life estima- tion for lithium-ion batteries using physics- informed neural networks.2024 IEEE Inter- national Conference on Prognostics and Health Management (ICPHM)(p. 67â 73). https://doi.org/10.1109/ICPHM61352 .2024.10627352 Qiang, B., Shi, K., Liu, N., Ren, J., Shi, Y.(2023).Integrating physics-informed recurrent Gaussian process regression into instance transfer for predicting tool wear in milling process.Journal of Manufactur- ing Systems,68, 42â55. https://doi.org/ 10.1016/j.jmsy.2023.02.019 Qiao, W., Liu, X., Huang, J., Wu, G. (2024). A prior knowledge embedding contrastive attention learning network for variable working conditions bearing fault diagnosis with small samples.IEEE Sensors Jour- nal,24(23), 39967â39980. https://doi.org/ 10.1109/JSEN.2024.3477456 Qin, L., Sun, T., Sun, X.-M., Xia, W. (2025). Managing battery performance degrada- tion using physics-informed learning scheme 74 for multiple health indicators.IEEE Transactions on Transportation Electrifi- cation. https://doi.org/10.1109/TTE.2025 .3525742 Qin, L., Zhang, S., Sun, T., Zhao, X. (2024). An interpretable neuro-dynamic scheme with feature-temporal attention for remaining useful life estimation.IEEE Transactions on Industrial Informatics,20(4), 5505â5516. https://doi.org/10.1109/TII.2023.3333933 Qin, Y., Liu, H., Wang, Y., Mao, Y.(2024). Inverse physicsâinformed neural networks for digital twinâbased bearing fault diagno- sis under imbalanced samples.Knowledge- Based Systems,292.https://doi.org/10 .1016/j.knosys.2024.111641 Qin, Y., Liu, X., Li, X., Mao, Y.(2025). Simulation-data driven generalized zero- shot learning for multi-agent bearing com- pound fault diagnosis.Knowledge-Based Systems,318.https://doi.org/10.1016/j .knosys.2025.113595 Raissi, M., Perdikaris, P., Karniadakis, G.E. (2019). Physics-informed neural networks: A deep learning framework for solving for- ward and inverse problems involving nonlin- ear partial differential equations.Journal of Computational physics,378, 686â707. https://doi.org/10.1016/j.jcp.2018.10.045 Ramirez, I., Pino, J., Pardo, D., Sanz, M., del Rio, L., Ortiz, A., . . . Aizpurua, J.I. (2024). Residual-based attention physics-informed neural networks for spatio-temporal age- ing assessment of transformers operated in renewable power plants.(Preprint at https://arxiv.org/abs/2405.06443v2) Ren, Y., Yi, R., Lian, Z., Xia, Q., Yang, D., Sun, B., Feng, Q. (2025). Health assessment of a brushless direct current motor stator using a physics-informed long short-term memory network.International Journal of Electrical Power and Energy Systems,164. https:// doi.org/10.1016/j.ijepes.2024.110402 Rizvi, S.H.M., & Abbas, M.(2023).From data to insight, enhancing structural health monitoring using physics-informed machine learning and advanced data collection methods.Engineering Research Express, 5(3), 032003. https://doi.org/10.1088/2631 -8695/acefae Russell, M., & Wang, P. (2022). Physics-informed deep learning for signal compression and reconstruction of big data in industrial con- dition monitoring.Mechanical Systems and Signal Processing,168. https://doi.org/10 .1016/j.ymssp.2021.108709 Sadoughi, M., Lu, H., Hu, C. (2019). A deep learn- ing approach for failure prognostics of rolling element bearings.2019 IEEE Interna- tional Conference on Prognostics and Health Management (ICPHM).https://doi.org/10 .1109/ICPHM.2019.8819442 Saha, B., & Goebel, K. (2007).Battery data set.(NASA Prognostics Data Repository, Moffett Field, CA) Sajedi, S., Eltouny, K.A., Liang, X. (2023). Twin models for high-resolution visual inspec- tions.Smart Struct. Syst,31, 351â363. https://doi.org/10.12989/s.2023.31.4.351 Sajjadi, P., Dinmohammadi, F., Shafiee, M. (2025).Machine learning in prognos- tics and system health management of cyber-physical systems: A review.IEEE Access.https://doi.org/10.1109/ACCESS .2025.3605553 Saxena, A., & Goebel, K. (2008).Turbofan engine degradation simulation data set.(NASA Prognostics Data Repository, Moffett Field, CA) SCImago(2024).SJRâSCImago Jour- nal & Country Rank.Retrieved from http://w.scimagojr.com(Accessed: September 25, 2024) Shang, J., Xu, D., Qiu, H., Gao, L., Jiang, C., Yi, P. (2024). A novel data augmentation frame- work for remaining useful life estimation with dense convolutional regression network. Journal of Manufacturing Systems,74, 30â 40. https://doi.org/10.1016/j.jmsy.2024.02 .011 Shi, J., Rivera, A., Wu, D.(2022).Battery health management using physics-informed machine learning: Online degradation mod- eling and remaining useful life prediction. Mechanical Systems and Signal Processing, 179. https://doi.org/10.1016/j.ymssp.2022 .109347 Singh, S., Ebongue, Y.E., Rezaei, S., Birke, K.P. (2023). Hybrid modeling of lithium-ion bat- tery: Physics-informed neural network for 75 battery state estimation.Batteries,9(6). https://doi.org/10.3390/batteries9060301 Singh, S.K., Khawale, R.P., Hazarika, S., Bhatt, A., Gainey, B., Lawler, B., Rai, R. (2024). Hybrid physics-infused 1D-CNN based deep learning framework for diesel engine fault diagnostics.Neural Computing and Applica- tions. https://doi.org/10.1007/s00521-024 -10055-y Song, M.-M., Xiong, Z.-C., Zhong, J.-H., Xiao, S.-G., Tang, Y.-H.(2022).Research on fault diagnosis method of planetary gearbox based on dynamic simulation and deep transfer learning.Scientific Reports, 12(1). https://doi.org/10.1038/s41598-022 -21339-5 Stoyanov, S., Sulthana, R., Tilford, T., Zhang, X., Hu, Y., Yang, X., . . . Wang, Y. (2025). Modelling the fatigue damage in power components using machine learning technol- ogy.Power Electronic Devices and Com- ponents,10. https://doi.org/10.1016/j.pedc .2025.100079 Su, Y., Shi, L., Zhou, K., Bai, G., Wang, Z. (2024). Knowledge-informed deep networks for robust fault diagnosis of rolling bear- ings.Reliability Engineering and System Safety,244. https://doi.org/10.1016/j.ress .2023.109863 Sun, B., Pan, J., Wu, Z., Xia, Q., Wang, Z., Ren, Y., . . . Feng, Q. (2023). Adaptive evolution enhanced physics-informed neu- ral networks for time-variant health prog- nosis of lithium-ion batteries.Journal of Power Sources,556.https://doi.org/10 .1016/j.jpowsour.2022.232432 Sun, G., Liu, Y., Liu, X. (2025). A method for esti- mating lithium-ion battery state of health based on physics-informed machine learning. Journal of Power Sources,627. https:// doi.org/10.1016/j.jpowsour.2024.235767 Sun, H., Cao, D., Zhao, Z., Kang, X. (2018). A hybrid approach to cutting tool remaining useful life prediction based on the Wiener process.IEEE Transactions on Reliability, 67(3), 1294â1303. https://doi.org/10.1109/ TR.2018.2831256 Sun, H., Peng, L., Lin, J., Wang, S., Zhao, W., Huang, S. (2022). Microcrack defect quan- tification using a focusing high-order SH guided wave EMAT: The physics-informed deep neural network GuwNet.IEEE Trans- actions on Industrial Informatics,18(5), 3235â3247.https://doi.org/10.1109/TII .2021.3105537 Sun, S., Peng, T., Zhou, Y., Zhang, X., Wang, D. (2024). Contrastive learning and dynam- ics embedding neural network for label-free interpretable machine fault diagnosis.ISA Transactions,144, 436â451.https://doi .org/10.1016/j.isatra.2023.11.018 Tang, J., Xiao, J., Chen, W., Li, X., Wei, C., Ding, X., Huang, W.(2024).A prior knowledge-enhanced self-supervised learn- ing framework using time-frequency invari- ance for machinery intelligent fault diagnosis with small samples.Engineering Applica- tions of Artificial Intelligence,133. https:// doi.org/10.1016/j.engappai.2024.108503 Tao, S., Zhang, M., Zhao, Z., Li, H., Ma, R., Che, Y., . . . Zhou, G.(2024).Non- destructive degradation pattern decoupling for ultra-early battery prototype verifica- tion using physics-informed machine learn- ing.(Preprint at https://arxiv.org/abs/ 2406.00276v1) Tefera, Y., Van Baelen, Q., Meire, M., Luca, S., Karsmakers, P.(2025).Constraint- guided learning of data-driven health indica- tor models: An application on the pronostia bearing dataset.(Preprint at https://arxiv .org/abs/2503.09113v1) Thelen, A., Lui, Y.H., Shen, S., Laflamme, S., Hu, S., Ye, H., Hu, C. (2022). Integrating physics-based modeling and machine learn- ing for degradation diagnostics of lithium- ion batteries.Energy Storage Materials,50, 668â695.https://doi.org/10.1016/j.ensm .2022.05.047 Tsui, K.L., Chen, N., Zhou, Q., Hai, Y., Wang, W.(2015).Prognostics and health management: A review on data driven approaches.Mathematical Problems in Engineering,2015(1), 793161. https://doi .org/10.1155/2015/793161 Tu, H., Moura, S., Wang, Y., Fang, H. (2023). Integrating physics-based modeling with machine learning for lithium-ion batteries. Applied Energy,329.https://doi.org/10 .1016/j.apenergy.2022.120289 Van De Schoot, R., De Bruin, J., Schram, R., Zahedi, P., De Boer, J., Weijdema, F., . . . 76 others (2021). An open source machine learning framework for efficient and trans- parent systematic reviews.Nature machine intelligence,3(2), 125â133. https://doi.org/ 10.1038/s42256-020-00287-7 Vogl, G.W., Weiss, B.A., Helu, M. (2019). A review of diagnostic and prognostic capabil- ities and best practices for manufacturing. Journal of Intelligent Manufacturing,30(1), 79â95. https://doi.org/10.1007/s10845-016 -1228-8 von Rueden, L., Mayer, S., Beckh, K., Georgiev, B., Giesselbach, S., Heese, R., . . . others (2021). Informed machine learningâa taxon- omy and survey of integrating prior knowl- edge into learning systems.IEEE Transac- tions on Knowledge and Data Engineering, 35(1), 614â633.https://doi.org/10.1109/ TKDE.2021.3079836 Wanasundara,S.N.,Wickramasinghe,A., Schaubroeck,M.,Muthukumarana,S. (2023).Detectingthermalanoma- lies in buildings using frequency and temporal domains analysis.Journal ofBuildingEngineering,75, 106923. https://doi.org/10.1016/j.jobe.2023.106923 Wang, A., Qin, P., Yuan, Y., Zhao, G., Sun, X. (2025). Physically informed hierarchical learning based soft sensing for aero-engine health management unit.Chinese Jour- nal of Aeronautics,38(3). https://doi.org/ 10.1016/j.cja.2024.11.018 Wang, B., Lei, Y., Li, N., Li, N. (2020). A hybrid prognostics approach for estimating remain- ing useful life of rolling element bearings. IEEE Transactions on Reliability,69(1), 401â412. https://doi.org/10.1109/TR.2018 .2882682 Wang, E., Zhou, H., Wen, G., Liu, Z., Chen, X. (2025). Koopman-informed neural network for machineryâs nonlinear dynamics learning and remaining useful life prediction.IEEE Transactions on Instrumentation and Mea- surement,74. https://doi.org/10.1109/TIM .2024.3509601 Wang, F., Wu, Z., Zhao, Z., Zhai, Z., Wang, C., Chen, X. (2024). Physical knowledge guided state of health estimation of lithium-ion battery with limited segment data.Relia- bility Engineering and System Safety,251. https://doi.org/10.1016/j.ress.2024.110325 Wang, F., Zhai, Z., Di, Y., Zhao, Z., Chen, X. (2025).Physics-informed neural network for satellite battery degradation estimation based on an extreme segment data within 2 min.IEEE Transactions on Industrial Elec- tronics. https://doi.org/10.1109/TIE.2025 .3546350 Wang, F., Zhai, Z., Zhao, Z., Di, Y., Chen, X. (2024).Physics-informed neural network for lithium-ion battery degradation stable modeling and prognosis.Nature Commu- nications,15(1). https://doi.org/10.1038/ s41467-024-48779-z Wang, F., Zhi, Q., Zhao, Z., Zhai, Z., Liu, Y., Xi, H., . . . Chen, X. (2023). Inherently interpretable physics-informed neural net- work for battery modeling and prognosis. IEEE Transactions on Neural Networks and Learning Systems, 1145â1159. https://doi .org/10.1109/TNNLS.2023.3329368 Wang, H., Liu, Z., Peng, D., Zuo, M.J. (2023). Interpretable convolutional neural network with multilayer wavelet for noise-robust machinery fault diagnosis.Mechanical Sys- tems and Signal Processing,195. https:// doi.org/10.1016/j.ymssp.2023.110314 Wang, L., Wang, F., Yang, Y., Liao, W.-H. (2025). A remaining useful life prediction approach for ball bearing by acoustic emission signal and physics-informed neural network.Struc- tural Health Monitoring. https://doi.org/ 10.1177/14759217251333056 Wang, L., Yang, T., Hu, B.(2025).A bat- tery state of health estimation method for real-world electric vehicles based on physics-informed neural networks.IEEE Sensors Journal. https://doi.org/10.1109/ JSEN.2025.3549486 Wang, Q., Wu, B., Zhu, P., Li, P., Zuo, W., Hu, Q. (2020). ECA-Net: Efficient chan- nel attention for deep convolutional neural networks.Proceedings of the IEEE/CVF conference on computer vision and pattern recognition(p. 11534â11542). https://doi .org/10.1109/CVPR42600.2020.01155 Wang, Q.B., Zhang, X.H., Lu, J.W., Xiao, G., Ren, Y.X., Li, W.J.(2025).Digi- tal twin-driven physically constrained gen- erative adversarial network for industrial boiler fault diagnosis.IEEE Transactions 77 on Instrumentation and Measurement,74. https://doi.org/10.1109/TIM.2025.3558806 Wang, S., Shuai, H., Hu, J., Zhang, J., Liu, S., Yuan, X., Liang, P.(2025).Few-shot fault diagnosis of axial piston pump based on prior knowledge-embedded meta learning vision transformer under variable operating conditions.Expert Systems with Applica- tions,269. https://doi.org/10.1016/j.eswa .2025.126452 Wang, S., Tao, J., Jiang, Q., Chen, W., Qin, C., Liu, C. (2024). A digital twin framework for anomaly detection in industrial robot system based on multiple physics-informed hybrid convolutional autoencoder.Jour- nal of Manufacturing Systems,77, 798â809. https://doi.org/10.1016/j.jmsy.2024.10.016 Wang, Y., Li, M., Zheng, L., Shi, M., Zheng, Z., Pei, X. (2024). Phyformer: A degradation physics-informed self-data driven approach to machinery prognostics.Advanced Engi- neering Informatics,62. https://doi.org/ 10.1016/j.aei.2024.102772 Wang, Y., Liu, S., Lv, S., Liu, G. (2025).Meta- learning and knowledge discovery based physics-informed neural network for remain- ing useful life prediction.(Preprint at https://arxiv.org/abs/2504.13797v1) Wang, Y.C., Tao, F., Zhang, M., Wang, L.H., Zuo, Y.(2021).Digital twin enhanced fault prediction for the autoclave with insuf- ficient data.Journal of Manufacturing Systems,60, 350â359. https://doi.org/10 .1016/j.jmsy.2021.05.015 Wang, Z., Zhou, Z., Xu, W., Sun, C., Yan, R. (2023). Physics informed neural net- works for fault severity identification of axial piston pumps.Journal of Manufacturing Systems,71, 421â437. https://doi.org/10 .1016/j.jmsy.2023.10.002 Weddle, P.J., Kim, S., Chen, B.-R., Yi, Z., Gasper, P., Colclasure, A.M., . . . Dufek, E.J. (2023). Battery state-of-health diag- nostics during fast cycling using physics- informed deep-learning.Journal of Power Sources,585.https://doi.org/10.1016/j .jpowsour.2023.233582 Wen, P., Ye, Z.-S., Li, Y., Chen, S., Xie, P., Zhao, S. (2023).Physics-informed neural networks for prognostics and health management of lithium-ion batteries.(Preprint at https:// arxiv.org/abs/2301.00776v2) Wu, W., Song, C., Zhao, J., Xu, Z.(2023). Physics-informed gated recurrent graph attention unit network for anomaly detec- tion in industrial cyber-physical systems. InformationSciences,629,618â633. https://doi.org/10.1016/j.ins.2023.01.136 Wu, Y., Liu, A., Huang, Z., Zhang, S., Van Gool, L. (2021, May). Neural architecture search as sparse supernet.Proceedings of the AAAI Conference on Artificial Intelligence, 35(12), 10379-10387.https://doi.org/10 .1609/aaai.v35i12.17243 Wu, Y., Sicard, B., Gadsden, S.A. (2024). Physics- informed machine learning: A comprehen- sive review on applications in anomaly detection and condition monitoring.Expert Systems with Applications,255. https:// doi.org/10.1016/j.eswa.2024.124678 Xian, W., Long, B., Li, M., Wang, H. (2013). Prognostics of lithium-ion batteries based on the verhulst model, particle swarm opti- mization and particle filter.IEEE Transac- tions on Instrumentation and Measurement, 63(1), 2â17. https://doi.org/10.1109/TIM .2013.2276473 Xie, F., Xiao, F., Tang, X., Luo, Y., Shen, H., Shi, Z. (2024). Degradation state assessment of IGBT module based on interpretable LSTM-AE modeling under changing work- ing conditions.IEEE Journal of Emerging and Selected Topics in Power Electronics, 1.https://doi.org/10.1109/JESTPE.2024 .3419042 Xie, W., & Zeng, Y. (2024). A knowledge distilla- tion based cross-modal learning framework for the lithium-ion battery state of health estimation.Complex and Intelligent Sys- tems,10(4), 5489â5511. https://doi.org/ 10.1007/s40747-024-01458-4 Xu, J., Wang, R., Liang, Z., Liu, P., Gao, J., Wang, Z.(2023).Physics-guided, data- refined fault root cause tracing framework for complex electromechanical system.Reli- ability Engineering and System Safety,236. https://doi.org/10.1016/j.ress.2023.109293 Xu, L., Deng, Z., Xie, Y., Lin, X., Hu, X. (2023). A novel hybrid physics-based and data-driven approach for degradation tra- jectory prediction in li-ion batteries.IEEE 78 Transactions on Transportation Electrifica- tion,9(2), 2628â2644. https://doi.org/10 .1109/TTE.2022.3212024 Xu, W., Wang, Z., Zhou, Z., Sun, C., Yan, R., Chen, X. (2024). Wear state assessment of external gear pump based on system-level hybrid digital twin.Mechanical Systems and Signal Processing,209. https://doi.org/10 .1016/j.ymssp.2024.111123 Xu, W., Zhou, Z., Li, T., Sun, C., Chen, X., Yan, R. (2024). Physics-constraint variational neural network for wear state assessment of external gear pump.IEEE Transactions on Neural Networks and Learning Systems, 35(5), 5996â6006. https://doi.org/10.1109/ TNNLS.2022.3213009 Xu, X., & Liu, C. (2024). Physics-guided deep learning for damage detection in CFRP com- posite structures.Composite Structures, 331. https://doi.org/10.1016/j.compstruct .2024.117889 Xu, Z., Guo, Y., Saleh, J.H.(2022).A physics-informed dynamic deep autoencoder for accurate state-of-health prediction of lithium-ion battery.Neural Computing and Applications,34(18), 15997â16017. https:// doi.org/10.1007/s00521-022-07291-5 Xu, Z., Zhao, K., Wang, J., Bashir, M. (2024). Physics-informed probabilistic deep network with interpretable mechanism for trustwor- thy mechanical fault diagnosis.Advanced Engineering Informatics,62. https://doi .org/10.1016/j.aei.2024.102806 Yan, B.X., Ma, X.B., Sun, Q.Z., Shen, L.J. (2024).Physics-enhanced NMF toward anomaly detection in rotating mechanical systems.IEEE Transactions on Reliability, 74(3), 3911-3925. https://doi.org/10.1109/ TR.2024.3417262 Yan, R., Zhou, Z., Shang, Z., Wang, Z., Hu, C., Li, Y., . . . Gao, R.X.(2025).Knowl- edge driven machine learning towards inter- pretable intelligent prognostics and health management: Review and case study.Chi- nese Journal of Mechanical Engineering (English Edition),38(1). https://doi.org/ 10.1186/s10033-024-01173-8 Yan, T., Wang, D., Wang, Y.(2024). Discrimination- and sparsity-driven weight- oriented optimization model for inter- pretable initial fault detection and fault diagnosis.IEEE Transactions on Instru- mentation and Measurement,73, 1â13. https://doi.org/10.1109/TIM.2023.3335512 Yan, T., Wang, D., Xia, T., Xi, L.(2024). Novel anchor discrimination learning for physics-informed machine degradation mod- eling.IEEE Transactions on Reliability, 73(1), 357â369.https://doi.org/10.1109/ TR.2023.3311769 Yang, Z., Xu, M., Wang, S., Li, J., Peng, Z., Jin, F., Yang, Y. (2024). Detection of wind turbine blade abnormalities through a deep learning model integrating VAE and neural ODE.Ocean Engineering,302. https:// doi.org/10.1016/j.oceaneng.2024.117689 Ye, J., Xie, Q., Lin, M., Wu, J. (2024). A method for estimating the state of health of lithium- ion batteries based on physics-informed neu- ral network.Energy,294. https://doi.org/ 10.1016/j.energy.2024.130828 Yin, C., Li, Y., Wang, Y., Dong, Y.(2025). Physics-guided degradation trajectory mod- eling for remaining useful life prediction of rolling bearings.Mechanical Systems and Signal Processing,224. https://doi.org/10 .1016/j.ymssp.2024.112192 Yin, Y., Tian, J., Liu, X. (2025, June). Remain- ing useful life prediction based on parallel multi-scale feature fusion network.Journal of Intelligent Manufacturing,36(5), 3111â 3127.https://doi.org/10.1007/s10845-024 -02399-y Yucesan, Y.A., & Viana, F. (2019). Wind turbine main bearing fatigue life estimation with physics-informed neural networks.Proceed- ings of the Annual Conference of the PHM Society 2019(Vol. 11). https://doi.org/10 .36001/phmconf.2019.v11i1.807 Yucesan, Y.A., & Viana, F. (2020). A hybrid model for wind turbine main bearing fatigue with uncertainty in grease observations. Proceedings of the Annual Conference of the PHM Society 2020(Vol. 12). https://doi .org/10.36001/phmconf.2020.v12i1.1139 Yucesan, Y.A., & Viana, F.(2021).Hybrid physics-informedneuralnetworksfor mainbearingfatigueprognosiswith visual grease inspection.Computers in Industry,125.https://doi.org/10.1016/ j.compind.2020.103386 79 Yucesan, Y.A., & Viana, F. (2022). A hybrid physics-informed neural network for main bearing fatigue prognosis under grease qual- ity variation.Mechanical Systems and Sig- nal Processing,171.https://doi.org/10 .1016/j.ymssp.2022.108875 Yucesan, Y.A., & Viana, F. (2023). Physics- informed digital twin for wind turbine main bearing fatigue: Quantifying uncertainty in grease degradation.Applied Soft Comput- ing,149.https://doi.org/10.1016/j.asoc .2023.110921 Zeng, L., Zhang, F., Lang, G., Wang, Y., Chen, Q. (2025). Application of frequency aware mechanism based physical information con- volutional neural network in rolling bear- ing fault diagnosis.IEEE Sensors Jour- nal.https://doi.org/10.1109/JSEN.2025 .3555423 Zgraggen, J., Guo, Y., Notaristefano, A., Huber, L.G. (2022). Physics informed deep learn- ing for tracker fault detection in photo- voltaic power plants.Proceedings of the Annual Conference of the PHM Society 2022(Vol. 14). https://doi.org/10.36001/ phmconf.2022.v14i1.3235 Zhang, L., Xia, B., Zhang, F. (2024). Adaptive fault detection for lithium-ion battery com- bining physical model-based observer and BiLSTMNN learning approach.Journal of Energy Storage,91. https://doi.org/10 .1016/j.est.2024.112067 Zhang, N., Jiang, Z., Sun, Y., Liu, Z., Hou, J., Wu, F. (2024). Model-data hybrid driven approach for remaining useful life prediction of cutting tool based on improved inverse Gaussian process.Journal of Manufacturing Processes,124, 604â620. https://doi.org/ 10.1016/j.jmapro.2024.06.027 Zhang, S., Liu, Z., Xu, Y., Chen, G., Su, H. (2025). An electrochemical aging-informed data-driven approach for health estima- tion of lithium-ion batteries with parameter inconsistency.IEEE Transactions on Power Electronics,40(5), 7354â7369. https://doi .org/10.1109/TPEL.2025.3532588 Zhang, S., Liu, Z., Xu, Y., Guo, J., Su, H.(2025).A physics-informed hybrid data-drivenapproachwithgenerative electrode-levelfeaturesforlithium-ion battery health prognostics.IEEE Trans- actions on Transportation Electrification, 11(1), 4857â4871. https://doi.org/10.1109/ TTE.2024.3471626 Zhang, T., Chen, J., Ye, Z., Liu, W., Tang, J. (2025).Prior knowledge-informed multi- task dynamic learning for few-shot machin- ery fault diagnosis.Expert Systems with Applications,271. https://doi.org/10.1016/ j.eswa.2025.126439 Zhang, Y., Feng, K., Ji, J.C., Yu, K., Ren, Z., Liu, Z.(2023).Dynamic model- assisted bearing remaining useful life pre- diction using the cross-domain transformer network.IEEE/ASME Transactions on Mechatronics,28(2), 1070â1080. https:// doi.org/10.1109/TMECH.2022.3218771 Zhang, Y., He, Y., Tang, H., Ren, Y., Xiang, J. (2024). Adversarial domain adaptation approach for axial piston pump fault diag- nosis under small sample condition based on measured and simulated signals.IEEE Transactions on Instrumentation and Mea- surement,73, 1â12.https://doi.org/10 .1109/TIM.2024.3385829 Zhang, Y., Wang, H., Shen, W., Peng, G. (2023).DuAK: Reinforcement learning- based knowledge graph reasoning for steel surface defect detection.IEEE Transactions on Automation Science and Engineering, 1â13.https://doi.org/10.1109/TASE.2023 .3307588 Zhang, Y., Wang, S., Zhang, C., Dui, H., Chen, R. (2025). Application of physics-informed machine learning in performance degrada- tion and RUL prediction of hydraulic piston pumps.Reliability Engineering and Sys- tem Safety,261. https://doi.org/10.1016/ j.ress.2025.111108 Zhang, Y., Zhang, M., Liu, W. (2025, Novem- ber 8). Joint distribution domain adapta- tion: a novel meta-learning framework for cross-domain few-shot fault diagnosis.Jour- nal of Intelligent Manufacturing. https:// doi.org/10.1007/s10845-025-02708-z Zhao, J., Feng, X., Pang, Q., Fowler, M., Lian, Y., Ouyang, M., Burke, A.F. (2024). Battery safety: Machine learning-based prognostics. Progress in Energy and Combustion Sci- ence,102. https://doi.org/10.1016/j.pecs .2023.101142 80 Zheng, Y., Chen, L., Bao, X., Zhao, F., Zhong, J., Wang, C. (2025). Prediction model opti- mization of gas turbine remaining useful life based on transfer learning and simultane- ous distillation pruning algorithm.Relia- bility Engineering and System Safety,253. https://doi.org/10.1016/j.ress.2024.110562 Zhong, J., Zheng, Y., Ruan, C., Chen, L., Bao, X., Lyu, L. (2025). M-IPISincNet: An explain- able multi-source physics-informed neural network based on improved SincNet for rolling bearings fault diagnosis.Informa- tion Fusion,115. https://doi.org/10.1016/ j.inffus.2024.102761 Zhou, M., Li, Y., Cao, Y., Ma, X., Xu, Z.(2025).Physics-informed spatio- temporal hybrid neural networks for pre- dicting remaining useful life in aircraft engine.Reliability Engineering and System Safety,256. https://doi.org/10.1016/j.ress .2024.110685 Zhou, Q., & Tang, J. (2023). Physics-informed machine learning with deep feature aggre- gation for bearing diagnosis.Proceedings of the International Congress on Sound and Vibration.Society of Acoustics. Zhou, Y., Zhang, Q., Huang, T., Cai, Z. (2024). Prior knowledge-augmented meta-learning for fine-grained fault diagnosis.IEEE Trans- actions on Industrial Informatics,20(6), 8115â8124.https://doi.org/10.1109/TII .2024.3367029 Zhou, Z., Li, T., Zhao, Z., Sun, C., Chen, X., Yan, R., Jia, J.(2023).Time-varying trajectory modeling via dynamic governing network for remaining useful life prediction. Mechanical Systems and Signal Processing, 182. https://doi.org/10.1016/j.ymssp.2022 .109610 Zhu, K., Guo, H., Li, S., Lin, X. (2024). Physics- informed deep learning for tool wear mon- itoring.IEEE Transactions on Industrial Informatics,20(1), 524â533.https://doi .org/10.1109/TII.2023.3268407 Zhu, K., Huang, C., Li, S., Lin, X. (2023). Physics- informed Gaussian process for tool wear prediction.ISA Transactions,143, 548â 556.https://doi.org/10.1016/j.isatra.2023 .09.007 Zhu, S.-P., Wang, L., Luo, C., Correia, J., de Jesus, A., Berto, F., Wang, Q.Y. (2023). Physics-informed machine learning and its structural integrity applications: State of the art.Philosophical Transactions of the Royal Society A: Mathematical, Phys- ical and Engineering Sciences,381(2260). https://doi.org/10.1098/rsta.2022.0406 Zhu, Y., Cheng, J., Liu, Z., Zou, X., Cheng, Q., Xu, H., . . . Tao, F. (2025). Data generation approach based on data model fusion: An application for rolling bearings fault diagno- sis with small samples.IEEE Transactions on Instrumentation and Measurement,74. https://doi.org/10.1109/TIM.2024.3504567 Zhu, Y., Cheng, J., Liu, Z., Zou, X., Wang, Z., Cheng, Q., . . . Tao, F. (2024). Remaining useful life prediction approach based on data model fusion: An application in rolling bear- ings.IEEE Sensors Journal,24(24), 42230â 42244. https://doi.org/10.1109/JSEN.2024 .3477489 Zhu, Y., Zi, Y., Li, J., Xu, J. (2024). Phys- iCausalNet: A causal- and physics-driven domain generalization network for cross- machine fault diagnosis of unseen domain. IEEE Transactions on Industrial Informat- ics,20(6), 8488â8498. https://doi.org/10 .1109/TII.2024.3369240 Zio, E. (2022). Prognostics and health manage- ment (PHM): Where are we and where do we (need to) go in theory and practice.Reli- ability Engineering & System Safety,218, 108119. https://doi.org/10.1016/j.ress.2021 .108119 Zjavka, L., MiËs Ěak, S., Prokop, L. (2017). NWP model revisions using polynomial similar- ity solutions of the general partial differen- tial equation.International Conference on Innovations in Bio-Inspired Computing and Applications(p. 81â91). https://doi.org/ 10.1007/978-3-319-76354-58 Zou, Q., Huang, P., When, S. (1996). Abrasive wear model for lubricated sliding contacts. Wear,196(1), 72-76.https://doi.org/10 .1016/0043-1648(95)06851-1 81 Appendix Excluded Studies After full-text analysis of the 212 eligible studies, 83 were excluded, most commonly due to a small number of recurring reasons. Many contributions focused on domains or assets outside the PHM scope adopted for this review (e.g., civil infrastructure or buildings), or did not perform any PHM task (e.g., quality-control or design-phase fatigue life prediction only). A substantial share of studies did not satisfy the operational definition of PIML used here: they were purely data-driven, relied exclusively on simu- lated data without real measurements, or used domain knowledge only in the form of conventional feature engineering. Additional studies were excluded because they did not employ ML at all, were abstract-only or doctoral-symposium contributions, or had been retracted. Table 21 details the excluded records (total- ing 83) and the rationale for their exclusion to ensure completeness and enhance the transparency of the review process. Table 21: This table lists all studies deemed outside the scope of this review following full-text analysis. For each study, the reference, the rationale for exclusion, and the reviewer are provided. Reviewers are identified as âAâ and âB,â corresponding to the two equally contributing authors (in no particular order). The rationale is not a summary of the respective work but concisely explains the basis for exclusion, i.e., some rationales may require consultation of the original study for full context. ReferenceRationaleRev. Carter, Imtiaz, and Naterer (2025) The use of abstract simulated systems (Tinkerbell attractor, R Ěossler attractor, and a continuous stirred tank reactor) leads to a scope mis- match, as the work is detached from industrial settings and lacks PHM relevance (see Sec. 3; Fig. 5). B Che et al. (2025)The âphysics extractorâ CNN is trained to learn the electrochemi- cal impedance spectroscopy from earlyQ-Vcurves, after which âthe learned physical features were augmented to the measured featuresâ for capacity prediction, representing standard deep learning (see Fig. 1; Sec. 4; Eq. 1). A S. Han, Awasthi, and Bollas (2025) The selection of a recursive model due to the dynamic nature of tool wear as well as the selection of input features based on domain knowledge is insufficient to qualify as PIML (see Sec. 4.5; Fig. 3). B Hong, Nie, Ji, and Ma (2025) A model is derived that can be used to quantify valve flow rate, but no PHM task is addressed, i.e., the work rather serves as a âreference for research on the design and control methods of hydraulic control systemsâ (see Sec. 5). B Kadiwala et al. (2025) The PDE is identified via sparse regression on the same empirical dataset and the âsolution of the PDE is then integrated into the fea- ture setâ for a standard Gaussian process regression model, i.e., feature engineering (see Sec. 2.3/2.4; Fig. 4). A B. Li, Zhu, and Zhao (2025) The proposed approach is designed to predict the fatigue life of alloy samples under multiaxial loading. Since the current state of the system is neglected by not taking live sensor data into account, this cannot be considered PHM (see Sec. 2.2.5; Fig. 4). B C. Li, Zhai, Fu, Qin, and Kang (2025) Alongside end-to-end feature extraction from raw data, time-, frequency-, and time-frequency-domain features are computed, mis- leadingly framed as leveraging prior knowledge, i.e., standard feature engineering (see Fig. 3; Sec. 3.1.2; Tab. 1). B Continued on next page 82 ReferenceRationaleRev. J.-X. Liao et al. (2025) Although labeled âphysics-informedâ, the method employs a conven- tional multi-term loss function that comprises kurtosis,l 2 /l 4 norm and cross-entropy, without incorporating any explicit physical laws or constraints (see Tab. 1; Sec. 3; Eq. 29). B F. Lu et al. (2025)Although they are integrated into the latent space of an AE, the prior information corresponds to common features (such as root mean square, standard deviation, kurtosis), which are therefore insufficient to qualify as PIML (see Sec. 3.1.1; Tab. 1). B Luo, Ju, Li, and Zhang (2025) Although the approach is physics-informedâusing a loss term to reg- ularize the monotonic relationship between membrane resistance and capacity lossâit is excluded by definition because the model is trained solely on simulated data (see Sec. 2.2; Fig. 4). A Stoyanov et al. (2025) By definition, using only so-called âphysics-informed datasetsâ gen- erated from high-fidelity thermo-mechanical finite-element simula- tions, without any real data, does not constitute PIML (see Fig. 1; Sec. 3/4.1). B G. Sun, Liu, and Liu (2025) This work is excluded as it was retracted at the request by the Editor- in-Chief due to plagiarism (seeA method for estimating lithium-ion battery state of health based on physics-informed machine learning, accessed on December 23, 2025). A Q.B. Wang et al. (2025) Since real-world data are used solely to validate the digital twin, and the proposed approach inherently relies on generated data to train the fault diagnosis model, this work is excluded by definition (see Fig. 9; Sec. I-D/IV-C). A S. Wang et al. (2025) The additional loss term designed to preserve distance information in the embedding space of continuous wavelet transformation snippets that are input to a vision Transformer cannot be considered as physically meaningful prior knowledge (see Sec. 3.2). B Y. Wang, Liu, Lv, and Liu (2025) The method assumes degradation follows an unknown PDE in a latent state space, inferring the governing dynamics from data, thereby learn- ing a PDE-like relationship between the hidden state and RUL without physical priors (see Sec. 3.2.3; Fig. 2). A T. Zhang, Chen, Ye, Liu, and Tang (2025) By learning ten âsignal feature indicatorsâ (such as max, min, stan- dard deviation) through an auxiliary task in a shared CNN, the method provides only feature self-supervision and does not incorporate prior physical knowledge (see Tab. 2; Sec. 2.4; Fig. 2). B Zheng et al. (2025) Although claiming to introduce prior knowledge, the method does not incorporate domain knowledge, relying instead on pre-trained weights and teacher outputs for transfer learning and distillation, i.e., a purely data-driven approach (see Sec. 3.1; Fig. 3). A Zhong et al. (2025) A âphysics-informed convolutional layerâ with analytically designed, fixed kernels based on bearing fault orders and Sinc-based bandpass filters is used as the first network layer, which effectively amounts to feature engineering (see Sec. 3.2; Tab. 1). B L. Chen et al. (2024) The âphysics-informedâ part fuses raw optical spectra (grayscale images) with a second input channel (selected emission lines, statistical/time-frequency features, and operating parameters), effec- tively performing feature engineering (see Sec. I-C; Fig. 6). B Continued on next page 83 ReferenceRationaleRev. C. Chen et al. (2024) The authors refer to their method as âinterpretable feature engi- neering,â relying on spectral denoising and extraction of the wheel perimeter-related dominant frequency as the interpretable feature, which does not qualify as PIML (see Sec. I-A/B; Fig. 8). B J. Chen et al. (2024) Although integrated into the latent space of a convolutional AE, the prior knowledge used solely consists of trivial statistical and frequency- domain features, which does not represent prior physical knowledge (see Sec. I; Fig. 1). B Exenberger, Di Salvo, Hirsch, Wotawa, and Schweiger (2024) Although the work is framed as relevant to predictive maintenance, it is limited to now-casting bearing temperature and does not perform any actual PHM tasks, leaving the connection to PHM superficial (see Sec. 1/2). A Fern Ěandez, Cor- betta, Kulkarni, Chiach ĚÄąo, and Chiach ĚÄąo (2024) Although technically representing a physics-informed approach, the work is excluded as it solely targets end-of-discharge prediction, where âfurther research should [be] undertaken about the aging effect in Li-ion batteriesâ (see Sec. 4). A Fern Ěandez, Chi- ach ĚÄąo, Barros, Chiach ĚÄąo, and Kulkarni (2024) While the proposed approach is indeed physics-informed, it specifi- cally addresses accelerations in concrete buildings under seismic events, thereby illustrating a scope mismatch (see Sec. 4.2). B Ge and Sadhu (2024) While physics-informed, the work falls outside the scope of this review, addressing structural health monitoring of civil infrastructure, i.e., steel beams and truss bridges (see Sec. 3/4). A G. Han et al. (2024) A CNN with a built-in wavelet feature-extraction layer and rein- forcement learning-guided selection is built, where âthe validation set accuracy [...] is taken as a priori knowledge,â obtained during pretrain- ing, making the approach purely data-driven (see Sec. I-A; Fig. 1; Eq. 9). A Jia et al. (2024)Extracting 17 statistical features from discharge, incremental capacity, and differential voltage curves, and augmenting them with degradation indicators from open-circuit voltage reconstruction and hybrid pulse power characterization testing, constitutes mere feature engineering (see Sec. 3; Fig. 4/5). A F. Jiang, Hou, and Xia (2024) While the approach may appear physics-informed, it assumes (without theoretical or domain-specific justification) a generic PDE on a learned latent state with empirically chosen derivative order, resulting in an arbitrary modeling choice (see Sec. 2.3/3.3). A Kayedpour et al. (2024) The proposed âhybrid physics-based deep learning frameworkâ for wind turbine diagnosis is solely trained on data generated from a multiphysics simulation and is therefore excluded from this review by definition (see Sec. I). B I. Kim et al. (2024) Using âprior knowledge that the bearing fault signals are impulse exci- tation signalsâ for signal processing, followed by a vanilla CNN and an explainable artificial intelligence technique, the method combines con- ventional feature engineering with a post-hoc explanation (see Fig. 2; Sec. 3.2). A Kumari and Wang (2024) By definition, training solely on simulated data, with no real-world data incorporatedâas exemplified here by a stochastic battery degradation modelâdoes not qualify as PIML (see Fig. 1). B Continued on next page 84 ReferenceRationaleRev. Lai, Baraldi, and Zio (2024) Due to the fact that the method for fault detection in electro-hydraulic servo actuators used in turbofan engine fuel systems involves solely sim- ulated data for the development of the reconstruction model, the work is excluded by definition (see Sec. 3.1). A T. Li et al. (2024)Since the data-driven components are limited to fixed low-order polyno- mial regressions identified from experimental and simulated data, with no explicit ML model being employed, the study ultimately lies outside the PIML scope (see Tab. 3). A Shang et al. (2024) By aligning run-to-failure sequences with dynamic time warping and averaging them via weighted barycenter to create additional time series, the approach is an interpolative, correlation-aware yet standard data augmentation method (see Fig. 2/4; Sec. 3.1). A Su, Shi, Zhou, Bai, and Wang (2024) Although labeled âknowledge-informedâ, the so-called knowledge-based features are merely standard time- and frequency-domain statistics, effectively amounting to conventional feature engineering (see Sec. 2.1; Fig. 2). B Tao et al. (2024)The proposed âultra-early prototype verification methodâ focuses on post-production quality assessment of lithium-ion batteries and is there- fore excluded, as it does not implement operational PHM (see Fig. 1; Discussions). A W. Xie and Zeng (2024) The teacher Transformer is pre-trained to learn degradation patterns and provide âprior knowledge of degradation lawsâ to the student CNNâknowledge that is merely learned from data (see Fig. 1/2; Methodology). A T. Yan, Wang, and Wang (2024) The method weights spectral lines using a generalized Rayleigh quotient eigenproblem, fuses each spectrum into a single âdegradation feature,â and classifies them by Euclidean distanceâreflecting classical linear algebra rather than ML (see Sec. I-C/D). A T. Yan, Wang, Xia, and Xi (2024) With the method being âformulated as a generalized Rayleigh quotient, which can be conveniently and easily solved by a maximum eigenvalue problem,â it exemplifies classical linear algebra rather than ML (see Sec. I-C; Eq. 11). B B.X. Yan, Ma, Sun, and Shen (2024) This study is excluded because it investigates high-speed train carriages, which are considered transportation systems and are therefore outside the scope of this review (see Fig. 1; Sec. VII). A Yang et al. (2024)The proposed method, embedding a vanillaNeural ODE(a purely data- driven model that learns dynamics from data) into an AE for anomaly detection, does not incorporate prior knowledge and is therefore not physics-informed (see Sec. 2.2/3.1.2). A Ye, Xie, Lin, and Wu (2024) Ambiguous âsecondary trainingâ protocols (testing/validation), alter- ing the physics constraint mid-work, and misreporting physical loss for the baseline neural network undermine the studyâs scientific rigor and comparability (see Sec. 2.3/3; Eq. 8/13; Fig. 8). A Y. Zhou, Zhang, Huang, and Cai (2024) While a âprior knowledge-augmentedâ meta-learning framework is pro- posed, it merely uses generic time-domain indicators as pseudolabels and heatmaps (gradient-weighted class activation mapping) fused at test time, not constituting a physics-informed approach (see Fig. 1/2; Tab. 1; Sec. I-A). A Continued on next page 85 ReferenceRationaleRev. Abadi (2023)While appearing in the proceedings of the annual PHM Society confer- ence, it constitutes a PhD research outline for the doctoral symposium track rather than a full technical paper, and is therefore excluded (see Sec. 3). A Cvijic, Gupta, and Lux (2023) Although the approach is basically suitable for diagnosis, prognosis, and RUL prediction of transformers, no quantitative results incorporating ground truth values are reported for the full framework (see Sec. 4). B Dwivedi, Yemula, and Pal (2023) Despite being labeled âphysics-inspired,â the method for detecting anomalous events is entirely data-driven, relying solely on âÎźPMU mea- surement data without any information about the network model or prior labeling of the eventsâ (see Abstract; Sec. I-B). B Fricke (2023)Even though it appears in the proceedings of the annual PHM Soci- ety conference, it represents a PhD research outline submitted to the doctoral symposium track rather than a full technical paper, and is therefore excluded (see Sec. 3). A Furlong and Reichard (2023) Although published in the proceedings of the annual PHM Society conference, it constitutes a PhD research outline submitted to the doc- toral symposium track rather than a technical paper, and is therefore excluded (see Sec. 3). A Garpelli et al. (2023) Although the approach embeds physics via a residual loss, it is excluded by definition because âsimulated data is used during the learning pro- cess of the neural network, while experimental data is employed for the testing phaseâ (see Conclusion). B Gij Ěon, Pujana- Goitia, Perea, Molina-Solana, and G Ěomez-Romero (2023) The approach targets wind turbine power modeling, explicitly framed as a âprevious step to developing optimal controllers and applying failure detection methods,â resulting in a scope mismatch, as no actual PHM tasks are performed (see Sec. 1). A Y. He et al. (2023)While presented as a ânovel physics-informed loss function,â the addi- tional term simply penalizes overestimation of RUL more than underes- timation for safety or economic reasons, which does not constitute prior physical knowledge (see Sec. 2.2). B He, Zhao, and Yan (2023) This approach is designed for offline fatigue life prediction, mapping from load parameters (stress/strain) to the cycle-based fatigue life. The current system health state is not considered. Therefore, this method doesnât qualify as PHM (see Sec. 1; Fig. 3). B Koutsoupakis, Seventekidis, and Giagopoulos (2023) By definition, the method does not qualify as PIML, since the CNN âis trained on numerical data alone,â generated via simulation for the purpose of identifying damage across different health states (see Sec. 5.1). B Kumar, Vetrive- lan, Kumba, and Ajeyprasaath (2023) Contrary to the claim of proposing a novel approach, âwhich integrates machine learning techniques with electrochemical modeling,â the imple- mented method is a conventional neural network, reflecting standard deep learning (see Sec. 2/4). A Lee, Kang, Kim, and Yoon (2023) While motivated by PHM, the work does not directly address PHM, as the fault detection and diagnosis methodology âthat uses the developed estimation model will be proposed in the future,â resulting in a scope mismatch (see Sec. 5). A Continued on next page 86 ReferenceRationaleRev. Lei et al. (2023)âEmbedding prior knowledgeâ is limited to computed order tracking-based data augmentationâresampling vibration signals via pseudo-speed ratios with amplitude scaling and Gaussian noiseâ followed by a standard metric-based meta-learner (see Sec. 3.1). A X. Liao, Chen, Wen, and Zhao (2023) With the underlying physics âcompletely unknown, a deepHPM can be used to approximate it,â the method effectively infers a PDE-like relationship between the hidden state and RUL from data, rather than integrating prior physical knowledge (see Sec. 2.2; Fig. 2). B Y. Liu, Wang, and Chu (2023) A method is proposed, âwhere the prior knowledge of machining param- eters is fused with features extracted from multiple sensor informationâ to construct âhybrid texture dataâ fed into a vision Transformer, i.e., feature engineering (see Sec. 2.2; Fig. 1/2). A Z. Ma, Liao, Gao, Nie, and Geng (2023) While both âphysics-informedâ feature extraction and parameter tuning are claimed, the former reduces to selecting leakage-related inputs and adopting rise time as the health indicator, and the latter is merely a vanilla hyperparameter search (see Sec. 2.1/2.2/3.2). A Mochammad, Noh, and Kim (2023) The work lacks key methodological details (e.g., low-fidelity model identification, datasets, training procedure, used loss function, and ques- tionable baseline comparisons), preventing a rigorous assessment of its contribution (see Sec. 2). B Tu, Moura, Wang, and Fang (2023) Although a series of hybrid models are proposed, with the sole focus of enabling highly accurate voltage prediction for lithium-batteries, the work does not directly address PHM and is therefore excluded (see Sec. 2.1; Fig. 2). A F. Wang et al. (2023) The approach is indeed PIML. As it is used for end-of-discharge predic- tion of lithium-ion batteries, hence not taking aging effects into account, it cannot be seen as PHM (see Sec. IV-F). B H. Wang, Liu, Peng, and Zuo (2023) The method is a standard supervised CNN trained with cross-entropy, merely embedding discrete wavelet transform and attention as signal- processing layers to improve feature learning and noise robustness (see Fig. 2; Sec. 3.5). A Weddle et al. (2023) By definition, training (in this case) VGG-16 solely on simulated bat- tery degradation does not qualify as PIML due to the absence of real-world measurements (see Sec. 3.4; Fig. 5). A W. Wu, Song, Zhao, and Xu (2023) Given that the âSecure Water Treatment testbed (SWaT) and Water Distribution testbed (WADI)â datasets are designed for cybersecu- rity research, the anomaly detection task does not target asset health, placing the work outside the scope of PHM (see Sec. 4.1). B Y. Zhang, Wang, Shen, and Peng (2023) While industrial, the work focuses on quality inspection of manufac- tured steel products rather than the health of industrial assets, which is the core concern of PHM, resulting in an inherent scope mismatch (see Sec. I/I-A). A Arias Chao, Kulka- rni, Goebel, and Fink (2022) While theoretically physics-informed, the use of âa discrete-time coun- terpart of the physics-based model F in the form of a deep neural networkâ effectively yields a purely data-driven approach (see Sec. 3.1; Eq. 5; Fig. 3). A Continued on next page 87 ReferenceRationaleRev. Y. Chen et al. (2022) The optimal parameters of the gearbox fault detection model considered in this study are determined by superimposing crack fault signatures onto validation data for hyperparameter optimization. This is a data augmentation technique (see Sec. 3.1; Fig. 3). B Hajiha, Liu, Lee, and Ramin (2022) The proposed âphysics-regularized data-driven approachâ relies, upon closer examination, exclusively on classical probabilistic modeling rather than traditional ML, revealing a scope mismatch (see Sec. 2). A Russell and Wang (2022) While stating that âincluding a loss term during AE training that is sensitive to frequency content introduces a physically informed objec- tive,â additional autocorrelation and Fast Fourier transform-based losses are effectively equivalent to standard multi-task learning (see Sec. 2.4). A Zgraggen, Guo, Notaristefano, and Huber (2022) Focusing on tracker faults that usually occur âwhen the tracker gets stuck at a certain orientation instead of tracking the sun,â this work addresses operational anomaly detection rather than asset health, and is therefore excluded (see Sec. 1). A Guo et al. (2021)In this approach, solely a N is used to estimate battery health indi- cators. A fractional-order model reconstructs measured quantities based on the health indicators for validation purposes, yet not directly relevant to health assessment (see Sec. 4.2/4.3). B Keizers, Loender- sloot, and Tinga (2021) Although the method aims to predict RUL, its reliance on an Unscented Kalman Filter for online parameter and state estimation rather than on ML leads to a fundamental mismatch in scope (see Sec. 3.3/3.4/4.3). A T. Li, Sbarufatti, Cadini, Chen, and Yuan (2021) The data-driven part is limited to an offline fitted, low-order polynomial measurement equation whose parameters remain fixed during progno- sis, resulting in a scope mismatch due to the lack of ML (see Sec. 3.2; Eq. 18). A Lyathakula and Yuan (2021) The work targets probabilistic fatigue life prediction based on offline test data, intended to replace/reduce expensive fatigue testing in the design phase of adhesively bonded joints, essentially not performing any PHM task using in-situ monitoring data (see Sec. 2). A Y.C. Wang, Tao, Zhang, Wang, and Zuo (2021) By definition, an approach that relies entirely on simulated data from a digital twin to train a CNN for autoclave fault prediction, without using actual operational data, does not qualify as PIML (see Sec. 6). B Cofre-Martel, Droguett, and Modarres (2020) This work is an abstract-only contribution and is therefore excluded for providing insufficient methodological and empirical detail (seeA physics-informed deep learning approach for fatigue crack propagation, accessed on December 23, 2025). B Kobrich, Mar- tin, Droguett, Bernardin, and Ayele (2020) Targeting crack growth prediction, the work proposes training a deep learning model solely on data simulated using extended finite element method, which by definition precludes it from being a PIML approach due to the absence of real-world measurements (see Sec. 4.2). A Akkad (2019)Although it appears in the proceedings of the annual PHM Society con- ference, it reflects a PhD research outline submitted to the doctoral symposium track rather than a full technical paper, and is therefore excluded (see Sec. 3). A Sadoughi, Lu, and Hu (2019) Using âphysics-based feature extraction, which is based on conventional signal processing techniques in time and frequency domainâ to obtain features such as root mean square, peak-to-peak, and kurtosis does not constitute a PIML approach (see Sec. I/I-B). A Continued on next page 88 ReferenceRationaleRev. Neerukatti, Chat- topadhyay, Iyyer, and Phan (2018) The proposed âhybrid prognosis modelâ for predicting crack propaga- tion is solely trained on data generated from finite element simulations, and is therefore excluded from this review by definition (see Prognosis model). A Z. Liu, Liu, Zhang, Cai, and Zheng (2015) The work focuses on a solar-assisted heat pump system and proposes a fault diagnosis method which is solely trained on incomplete simulation data, and is therefore excluded by definition (see Abstract; Sec. 5). A Kulkarni, Biswas, Celaya, and Goebel (2013) The development of physics-based degradation models is proposed to predict electrolytic capacitor aging under thermal overstress, using experimental data for calibration without employing any ML techniques (see Sec. 3.2; Fig. 2). A 89